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Models for tent caterpillar-virus interactions Beukema, Sarah Jenelle 1992

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MODELS FOR TENT CATERPILLAR-VIRUS INTERACTIONSbySarah Jenelle BeukemaB.A. Kalamazoo College, 1988A thesis submitted in partial fulfillment ofthe requirements for the degree ofMaster of ScienceinThe Faculty of Graduate StudiesDepartment of ZoologyandInstitute of Applied MathWe accept this thesis as conformingto the required standardThe University of British ColumbiaApril 1992© Sarah Jenelle Beukema, 1992In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.Department of ^The University of British ColumbiaVancouver, CanadaDate ^DE-6 (2/88)ABSTRACTMany species of forest Lepidoptera show eight to twelve year population cycles whichmay involve viral disease. To examine possible interactions between viral disease andpopulation cycles of forest Lepidoptera I explored some models for insect-virus dynam-ics. All of the models produced population oscillations in their original form. However,after they were modified to conform more closely to the tent caterpillar system, noneof the models produced realistic cycles. I then developed a new model specifically forthe tent caterpillar system that included important features such as: reduced fecundityof individuals that had been exposed to the disease, transmission of the disease frommother to progeny, a free-living infective stage of the virus, and a horizontal transmis-sion rate that varied with the larval stage, the number of individuals, and amount ofvirus present. Cyclic dynamics resulted from some simulations. The parameters pro-ducing the cycles were similar to actual data. However, unlike natural populations oftent caterpillars, in the simulated population the average fecundity decreased before thepopulation started to decline and survival decreased at approximately the same time asthe population. Important further research in the field should include investigation ofthe distribution and survival of free-living virus and factors that would reduce caterpillarsurvival at peak populations but not affect fecundity.TABLE OF CONTENTSABSTRACT ^  iiTABLE OF CONTENTS ^  iiiTABLES ^FIGURES  viACKNOWLEDGEMENTS ^  vii1. INTRODUCTION  11.1 Overview ^  11.2 Organization  32. BIOLOGY OF TENT CATERPILLARS ^  43. MODELS ^  83.1 What is a model? ^  83.2 Modelling Background  93.2.1 Exponential and logistic growth ^  93.2.2 Lotka & Volterra ^  113.2.3 Kermack McKendrick ^  143.2.4 Nicholson & Bailey and beyond ^  163.3 Models that produce cycles  173.3.1 Predator-prey ^  183.3.2 Weather  203.3.3 Plant quality  213.3.4 Parasites ^  233.3.5 Diseases: systems with immunity ^  233.3.6 Diseases: systems with no immunity  243.3.7 Disease and genetics ^  254. INSECT-VIRUS MODELS  274.1 Introduction ^  274.2 Anderson and May  284.3 Brown ^  324.4 Regniere  364.4.1 Original model ^  364.4.2 Modifications  394.5 Diekmann and Kretzchmar ^  565. A NEW MODEL ^  605.1 Introduction  605.2 Formulation of the model ^  615.2.1 Model overview  615.2.2 Larval stage ^  645.2.3 Pupal stage  675.2.4 Adult stage  685.3 Parameter values ^  72nl6. SIMULATION RESULTS AND ANALYSIS ^  776.1 Introduction ^  776.2 Simulation results  786.3 Analysis  906.3.1 Introduction ^  906.3.2 Virus reproduction rate ^  916.3.3 Stability 926.3.4 Comparison with simulation results ^  956.4 Summary ^  977. CONCLUSIONS  998. LITERATURE CITED ^  106Al. APPENDIX 1: ANALYSIS OF MODELS ^  110A1.1 Introduction  110A1.2 Lotka-Volterra ^  110A1.3 Nicholson-Bailey  111A2. APPENDIX 2: ANALYSIS OF THE NEW MODEL ^ 113A2.1 Introduction ^  113A2.2 Virus reproductive rate ^  113A2.3 Equilibrium analysis  114A2.4 Stability: numerical analysis  118ivTABLESTable 1. Effect of increasing parameters in the modified R,egniere models. . . . 43Table 2. Summary of the different forms of p t used in the modified Regnieremodel. ^  49Table 3. Parameter estimations for b and c based on data^  54Table 4. Variables and parameters used in the new model  63Table 5. Parameter values and effect of increasing them in the new model. .^79vFIGURESFigure 1. The basic life cycle of the tent caterpillar^  5Figure 2. Effect of increasing r on the behaviour of the discrete logistic equation. 12Figure 3. Predator-prey cycles from the Lotka-Volterra model. ^ 19Figure 4. Output from Brown's original model^  34Figure 5. A schematic diagram of Regniere's model  37Figure 6. Existence and period of cycles in the modified Regniere model. . . . ^ 41Figure 7. Effect of varying net fecundity of virus-exposed individuals on equations4.7.  ^44Figure 8. Existence and period of cycles in the modified Regniere model which^includes free-living virus.   46Figure 9. A comparison of the different equations for the infection probability thatwere used in equations 4.11. ^  50Figure 10. A comparison of the different forms between birth terms and for theinfection rate 0   59Figure 11. Lifecycle of the tent caterpillar with the corresponding parameters andvariables from the model. ^  62Figure 12. Existence and period of cycles in the model as the parameters /3, cry , Land f e are varied.^ 81Figure 13. Maximum and minimum clutch sizes produced during each of the cyclesindicated in Figure 14.   84Figure 14. Log number of tents and average clutch size in two actual populationsand a simulated population. ^  85Figure 15. Early survival rates (from egg to fourth instar) in a simulated and anatural population    87Figure 16. A comparison between various forms for tot , the amount of virus ingestedat each instar. ^  88Figure 17. Constant p vs equilibrium number of susceptibles^  94Figure 18. Parameters x and z from the analysis vs constant p  117viACKNOWLEDGEMENTSI would like to thank Don Ludwig and Judy Myers for always being there to guide meand for their helpful comments at all stages of the work. I also deeply appreciate theencouragement and support from Gordon Davidson and Andrew Trites.vii1. INTRODUCTION1.1 OverviewMany forest lepidopteran species show eight to twelve year population cycles (Myers,1988). At peak densities, the larvae often defoliate trees and are considered to be severepests. Tent caterpillars are a typical cyclic species. Viral disease is usually associatedwith decline populations of cyclic caterpillars and may be involved in the populationdynamics Numerous models for virus-insect interactions have been used to explorepossible dynamics between virus and its hosts (Anderson and May, 1980; Brown, 1987;Regnieie, 1984; Diekmann and Kretzchmar, 1991). To investigate the applicability ofthese models to a realistic field situation, I adapted several of them to details of thetent caterpillar system. All of the models produced population oscillations (usually inthe form of limit cycles) in their original form. However, after they were modified toconform more closely to the tent caterpillar system by adding vertical transmission ofthe virus, lower fecundity of exposed insects and a free-living virus stage, and simulatedusing realistic parameter values, none of the models produced reasonable cycles.Even though no previous insect-virus model produces reasonable cycles, it is stillpossible that virus creates the natural population cycles. To examine this, I createda new model specifically for the tent caterpillar system. It included various importantfeatures such as: reduced fecundity of individuals that had been exposed to the disease,transmission of the disease from mother to progeny, free-living infective stage of thevirus, and a horizontal transmission rate that varied throughout the larval stage andthat depended on the number of individuals and amount of virus present.1The model was simulated using a wide variety of parameter values and analysiswas done on a simplified version of the model. Although the analysis showed thatthe simplified model would not produce cycles, eight to twelve year oscillations didoccur with some parameter values in the full model. However, in all cycles that areproduced, the patterns shown in the oscillations are different than those seen in naturalpopulations. Natural populations show that the average clutch size remains high even inthe first year of a population decline while survival rates decrease before the populationdecline. In the model, the clutch size is reduced before the population decreases, andis the cause of the population decline. Survival rates decrease at the same time as thepopulation. Thus, on the basis of current belief about the interactions between virusand insects and about the biology of the virus, it is unlikely that virus is the sole causeof cycles. However, it is possible that virus combines with some other factor to createthe cycles.21.2 OrganizationThe thesis is organized into several parts. In the Chapter 2 I describe the biology oftent caterpillars. Chapter 3 contains background information about models including adefinition of a model, a description of how models can be used in biology, a brief accountof some of the important models in biology and a discussion of various hypotheses andmodels that produce cycles. In Chapter 4 insect-virus models are examined in moredetail, and various modifications to the models are described. Finally, a new modelcreated for tent caterpillar-virus interactions is formulated in Chapter 5 and the resultsof simulations and analysis are discussed in Chapter 6. All the actual analyses of thevarious models are in Appendices Al and A2.32. BIOLOGY OF TENT CATERPILLARSAt least sixteen species of forest lepidoptera in the North Temperate zone show popu-lation cycles. Some well studied examples are the spruce budworm, the tussock mothand the larch budworm, which reach outbreak levels every eight to twelve years andperiodically are major defoliators of trees (Myers, 1988). Tent caterpillars, Malacosomaspp, also show cyclic dynamics and the western and forest tent caterpillars have beenextensively studied (Daniel, 1990; Myers, 1990).Tent caterpillars are univoltine. The adult female lays her eggs in a single egg massin July and then dies. Neonate caterpillars overwinter in the eggs and hatch early thefollowing spring. Caterpillars from a single egg mass remain together and form silk tentsin trees. The size of the tents is determined by the age and number of the caterpillarsin the group. As a consequence, the size and number of tents are good indicators ofpopulation size (Myers, 1990). Larvae feed gregariously until the fifth instar when theymay start wandering individually and may move more than thirteen meters from theirtent before pupation 50 to 80 days after hatching (Wellington et al., 1975). Adult mothsmate shortly after emergence and females lay their eggs soon after mating. The basiclife cycle is shown in Figure 1.Many agents kill caterpillars, including a variety of parasitiods and predatory in-sects (Smith and Goyer, 1986). Caterpillars can also be infected by a nuclear poly-hedrosis virus, bacterial and fungal diseases. Much work has been done on differentnuclear polyhedrosis viruses of Lepidoptera since they are usually species specific andhave great potential as biological control agents (Entwistle, 1983a).4reproduce,dieEggs overwinter(July - April) Virus^Larvae(Instars 1-5)(April - June)Adults(July )Pupae(June - July)Figure 1. The basic life cycle of the tent caterpillar.5Nuclear polyhedral virus (NPV) can be transmitted in two ways. Most common istransmission within a generation (horizontal transmission). As early as eight days afterinfection, the infected larva dies (Clark, 1958). The corpse disintegrates rapidly and thevirus propagules (polyherdral inclusion bodies, PIBS) are released onto the surroundingleaf. Other caterpillars become infected by ingesting the virus while feeding (Clark,1955; Entwistle, 1983a). Caterpillars can also spread the disease by moving over thecontaminated leaves (Stairs, 1965). The larger (older) the caterpillar, the more viralpropagules are produced when the organism dies. However, larger larvae are also lesssusceptible to the disease, so that more PIBS must be ingested to initiate infection(Entwistle, 1983b).NPV is also apparently transmitted from a mother to her progeny (vertical trans-mission) (Clark, 1955, 1958; Myers, 1990; Wellington, 1962), most likely on the coatingsurrounding the egg. A portion of the caterpillars will become infected at hatchingif they ingest viral polyhedra. If eggs laid by females from a virus infested area arebrought into a disease-free laborabory or field site, the larvae arising from these eggswill die from NPV infections (Clark, 1955, 1958).Whether a pool of virus exists on the food plants or in the soil is debated (Clark,1956; Entwistle, 1983a). If such a reserve exists, it could also be a means of infectionbetween generations. Clark (1956) showed that caterpillars arising from eggs from adisease-free area but that had been transferred to a site with a previous population ofdiseased individuals showed viral infection, indicating that there is a free-living viruspool and that the virus can survive over the winter.6Many features of the relationship between the larvae and the virus are not yetwell understood. Can a sublethally infected caterpillar (or a caterpillar that is exposedto the virus during the late larval stage but does not die from the virus) act as acarrier for the virus and pass it on to her offspring? If so, what proportion of viraltransmission occurs in this way? Laboratory experiments using another lepidopteran,Spodoptera ornithogalli (yellow striped armyworm), have shown that although exposureto the disease (through the larvae eating food infected with the virus PIB's) is notsufficient for vertical transmission, vertical transmission can occur via adult moths thatare contaminated with virus. The larvae that emerge from the eggs laid by these infectedmoths can become infected (Young, 1990). Tent caterpillars from eggs collected fromdeclining populations die from viral and fungal disease when reared in the laboratory(Myers, pers. comm. and 1990; Clark, 1955).Regardless of how the disease is transmitted vertically, some larvae are born infected(or become infected as they are emerging from the egg). This leads to further questionssuch as if an organism is born infected, will it survive to reproductive age and if itdoes, will it be able to reproduce? For the yellow striped armyworm mentioned above,the larvae can be exposed to the disease (from eating food with PIB's) and not die.However the adults may suffer reduced fecundity as a consequence (Young, 1990). It ispossible the same holds for tent caterpillars. Even though these questions remain to beanswered, effects of such assumptions can be explored using models.73. MODELS3.1 What is a Model?A model is a means of describing something using words, objects and/or equations. Inthe physical sciences, models are usually considered to be quite reliable since people areusually concerned with relatively deterministic and well understood events. However,in biology events are much more variable, and there tend to be fewer data to workfrom. Therefore, models in biology are often more speculative, but can be used toclarify ideas or to explore the predictions of a hypothesis (Starfield and Bleloch, 1986).Models can also suggest what data should be collected or what experiments should beperformed in the field to test hypotheses. A 'working' model is often considered tobe one in which there is agreement between observations and the model output. It isimportant to realize that in biology, a 'working' model does not necessarily mimic theway nature operates: often there are many different models that give the same results.For example, as will be seen, cycles can arise from a wide variety of different hypotheses.If one model produces a desired result, all that can be concluded is that the hypothesisthat the model was based on is still feasible. If a model does not produce output whichcorresponds to natural occurances, it indicates that the hypothesized interactions areincomplete, the functions to describe these interactions are inappropriate or that analternate hypothesis should be formulated. Consequently, more is often learned from amodel that does not produce an expected outcome.The first section of this chapter describes some of the classic models in biology,namely the exponential and logistic models, the Lotka-Volterra models for predator-8prey interactions, the Kermack-McKendrick epidemiology model and the Nicholson-Bailey model for host-parasitoid interactions. Each of these models is still used todayas a basis for creating more detailed models. In the second section, I discuss existinghypotheses and models for population cycles.3.2 Modelling Background3.2.2 Exponential and Logistic Growth.While mathematical models have only recently been widely used in biology, the basicideas of simple population growth extend back to the eighteenth century when interestin human demography led to calculations of the doubling time and carrying capacity forthe human population (Kingsland, 1985; Hutchinson, 1978). Although he was not thefirst, T.R. Malthus is usually credited with the realization that populations will growgeometrically when unchecked in any way (Hutchinson, 1978; Malthus, 1798). Themodel for this basic population growth can be formulated by stating the tautology thatthe rate of change in the size of a population depends on the births minus the deaths:in mathematical termsdN^dt = bN — dN^(3.1)where b and d are constant birth and death rates. If b and d are independent of N andt, equation 3.1 can be solved to give:^N(t) = No e(b—d)t .^ (3.2)This exponential growth model is often labelled as the Malthusian model, and theparameter r = b— d is sometimes referred to as the Malthusian parameter (Hutchinson,91978). Although the model was originally formulated as a model for human populationgrowth, it is now generally used only in the study of population growth in bacteria,yeast and other unicellular organisms.No population can continue to grow indefinitely in exponential mode. Eventuallysome factor, such as lack of space or food, will limit this growth. In 1836, Verhulst firstproposed the logistic model which includes these ideas:dN= r(1 - -k-)Ndt(Hutchinson, 1978). The birthrate is the term r(1-N/k) where k is the carrying capacityfor the population (the maximum population size the environment can sustain, due tospace or food or some other limiting resource) (Murray, 1989). This model was alsoderived independently by several researchers (including A.J. Lotka and R. Pearl) in the1920's, and popularized at that time by Pearl (Kingsland, 1985). Consequently, thelogistic model is sometimes referred to as the Pearl-Verhulst equation.Upon examination of the model, it is clear that if the population N is small, thebirthrate term will be close to r and equation 3.3 will behave essentially like equation 3.1and exhibit exponential growth. However, when the population is near k, the birthrateterm will be very small. Equation 3.3 may be solved in the formNo kertN(t) = k No(ert 1)•(3.4)It can be seen that as t^oo, N -p k. If the population is larger than k, the birthratewill be negative and the population will decline. So for any initial population size andparameter values, the population will always approach k as t^oo.10(3.3)There is also a discrete version of the logistic equation:NNt+1 = Nt + r(1 — tT )Nt (3.5)(May, 1973). This model behaves very differently from the continuous model. Figure2 shows that, as the value of r increases, the model behavior changes. For low valuesof r, the population will stabilize. If r is increased, the model will show cycles thatdecay down to some stable level. A larger r produces stable two-point cycles. Thecycle period increases as r increases until eventually the population behaviour is chaotic(May, 1973).3.2.3 Lotka & VolterraThe next major breakthrough in modelling came almost simultaneously from a physicalchemist, A.J. Lotka, and a mathematical physicist, V. Volterra in the 1920's (Kingsland,1985). They each proposed a system of equations to model the interactions betweentwo populations. Lotka derived the model as an extension to his work with singlespecies populations. Volterra wanted to explain the oscillations in the catch levels ofcertain species of fish by using predator-prey interactions (Volterra, 1926). He usedbasic balance equations involving the rate of change of the prey population N and thepredator population P, and made the following assumptions:1. Each population is made up of organisms that are exactly the same as each other,(for example, in age and behaviour), and this composition of the population doesnot vary over time.2. Prey would increase exponentially at a rate r without predators present and preda-tors would decrease exponentially at a rate d with no prey present.110 10^20^300I^1   ^I^1^1^1^1^1^1^1\ , \ \ \^I 1 \ 4B) r=2.1TimeFigure 2. Effect of increasing r on the behaviour the discrete logistic model. Thefirst graph show the population stabilizing. The next graphs show two-point cycles,four-point cycles, and finally, chaos.123. The encounter rate between prey and predators is governed by mass action. Thus,predation reduces prey growth at a rate proportional to the predator and preypopulations bN P (a different b from Section 3.2.2, equations 3.1 and 3.2).4. Predator growth due to prey is also a rate proportional to the size of both popula-tions cNP.5. Predators and prey population sizes react instantaneously to changes.(Volterra, 1926; Kingsland, 1985; Murray, 1989). Thus the system of equations is:dN = rN — bNP= cNP — dP. (3.6b)These equations are usually called the Lotka-Volterra model. The analysis of this modelis shown in Appendix 1.2, where it can be seen that the populations will always cycle.The model is not useful in modelling real populations since the predicted populationbehaviour (the period and amplitude of the cycles) is entirely dependent on the initialconditions and on the parameter values, none of which can ever be estimated exactly.However, the model is important as a basis for building other, perhaps more realistic,predator-prey models.dtdPdt(3.6a)133.2.4 Kermack Si McKendrickSoon after Volterra proposed his model, two other classic models were created. Thefirst of these was an epidemic model by Kermack and McKendrick (1927). At thattime, there was much controversy about what caused the termination of an epidemic.The two theories that were popular were that: a) there were no more susceptible peopleand b) the virulence of the disease had decreased during the epidemic (Kermack andMcKendrick, 1927). Kermack and McKendrick decided to use a simple model to examinethe question. They looked at the effects of disease in a closed population (i.e., nobirths, deaths or other movement in or out of the population), in which the populationwas divided into three classes: susceptibles S, infecteds I and removed R. Contactwith an infected individual is necessary for transmission of the disease to occur, so thesusceptibles become infected at a rate which is proportional to the number of infectedsand susceptibles individuals present in the population (0/5). Thus,dSat OIS. (3.7a)The infected individuals are removed from the population (either by immunity or death)at a rate v. So the equations for the final two classes aredIdt OIS — vIdR = vI.di(3.7b)(3.7c)Through analysis of this model, Kermack and McKendrick determined that the epidemicdies out before the susceptible population is exhausted. There is some population14threshold level that can be easily derived. First, rewrite equation 3.7b asdITit = (s - ')iLetVNT=.Assume that the initial conditions for the model state that there is a large amount ofsusceptibles (So ) present, and few infecteds (Is ). Then, according to equation 3.8, therate of increase of the infecteds will only be positive if S o > NT (Anderson, 1991). Therate of increase of infecteds must be positive for an epidemic to occur. If the populationis smaller than NT, the infecteds will eventually be removed, and an epidemic cannottake place. This result is referred to as the "threshold theorem". This theorem and thismodel are the foundations for much of subsequent epidemiology modelling (Anderson,1991).(3.8)(3.9)153.2.5 Nicholson & Bailey and beyondThe entomologist, A.J. Nicholson and physicist, V.A. Bailey proposed the second of theclassic models in 1935. Nicholson originally wanted to examine possible hypotheses ofpopulation regulation, and started with host-parasite interactions (Kingsland, 1985).Bailey assisted him with the mathematics. Initially, they examined host-parasitoidinteractions in which the parasitoid kills the host. In their model, they assumed thatonly the fraction of unparasitized hosts can reproduce. The number of parasites (P)depends on reproductive rate of the parasitiods (c) in the parasitized hosts (N(1 — f)).Assuming random encounters, the fraction that escapes parasitism (f) can be expressedusing a Poisson distribution (Edelstein-Keshet, 1986),^f = e—aPt^(3.10)and the final equations are:Nt+1^AN t a Pt^(3.11a)Pt+1 = CAri (1 —^e—aPt (3.11b)This model is analyzed in Appendix 1.3, and is shown to produce growing, unstable,oscillations. This model has been the basis for a large number of other parasite models(see Edelstein-Keshet (1986) for a brief description of some of the model's modifications).In the late 1960's and early 1970's modelling in population ecology became quitepopular. There were several books written at that time (Pielou, 1969; May, 1973)which discussed several of the basic models, their properties, applications, and somemodifications. Since then, modelling has become increasingly popular, both among16mathematicians and biologists. As computers become faster and more powerful, modelsare becoming more complex and analytically intractable. But the basic purpose remainsthe same: to use models as an effort to clarify and to understand what is happening innature.3.3 Models That Produce CyclesThere are many hypotheses about potential causes of population cycles. Some of theseinclude predator-prey interactions, weather patterns, plant quality, parasites and dis-eases. In all cases, the hypotheses contain the assumption that it is just one majorfactor that is driving the cycles. Finerty (1980) discusses the existence of populationcycles in small mammals and the relevance of many of these hypotheses to the cycles.Models, both theoretical and simulation, have been created for each of these hypotheses,and many show cycles. In this section, I discuss existing models for population cyclesof organisms in general, not just those of insects, and mention some of the different hy-potheses for population cycles and some of the major models created to support thesehypotheses. The relevance of some of these hypotheses to the tent caterpillar system isalso discussed.173.3.1 Predator-preyThe predator-prey hypothesis states that as the prey numbers increase, the predatornumbers will also increase until such a point when the predators actually reduce thegrowth rate and number of prey. This causes the predator numbers to decrease fromlack of food. When the predators become few enough, the prey population can start togrow again. Figure 3 gives an example of predator-prey cycles.The classic Lotka-Volterra model (see equations 3.4) for predator-prey interactionsshow these cycles. However these are neutral cycles, meaning that the period andamplitude of the cycle depend on the parameter values and the size of the startingpopulation. The model is structurally unstable since small changes in parameters givedifferent results. If a researcher were trying to use the Lotka-Volterra model to predictbehaviour of the population, and if the parameter estimates were even slightly differentfrom the true parameter values, the predicted period and amplitude of the cycle wouldbe different. Thus the model is impractical, since parameters such as reproductive andmortality rates vary from year to year and population to population. This model isuseful as a starting point for creating other predator-prey models (or plant-herbivoremodels). However, predation does not seem to be a major factor in the tent caterpillarsystem and it is more likely that something else is driving the cycles.18I^I^I30 40i20Generalions13i3Figure 3. Predator-prey cycles193.3.2 WeatherWeather variation has been hypothesized as a control of insect outbreaks (Martinat,1987). Bad weather will tend to keep insect populations at a low level (Andrewarthaand Birch, 1954). If there is a period of weather that is beneficial to the insects' survivalor fecundity for several generations, then the insects may be able to increase rapidly(Andrewartha and Birch, 1954). Weather can have both direct and indirect effects oninsects (Martinat, 1987). Direct effects act on the insect without any intermediaryagent and include the effects of weather on the behaviour or physiology of the insect(Martinat, 1987). For example, the development rates of many insects are connectedto the temperature. Indirect effects are those in which the weather affects food sourcesor predators (including parasites) of the insect. Such a case could be that the effect ofbad weather may cause a plant to become stressed and to produce fewer defenses or toalter in nutritional value. If the plants had fewer defenses, insects would be able to eatmore of them, and the insects' survival would increase (Martinat, 1987).Weather patterns were included as a critical factor in the simulation models of tentcaterpillars by Wellington et al. (1975; Thompson et al., 1979). Growth, mortality,disease, and dispersal were all simlated to depend on the weather patterns of the area.They also included other aspects of the biology such as parasitism and the condition ofthe larvae. The weather patterns that they used initially in the model were all basedon available data about the actual weather in the study area. In their model, they wereable to produce population trends that corresponded to their data. It seems possiblethat the weather was driving the behaviour of the system, especially since when they20altered the weather patterns in the model, some of the results changed and no longermatched the behaviour of the natural system.In other tent caterpillar systems however, the cycles do not seem to be relatedto the weather (Daniel, 1990; Myers, 1981). After examining maps and data, Daniel(1990) concluded that larval feeding temperature and overwintering temperature do notexplain the dynamics of the forest tent caterpillar in Ontario, although the temperaturesmay determine the long-term susceptibility of a particular area to population outbreaks.Myers (1981) also noted that since in her populations, rainfall often can be correlatedwith the increase phase of the cycle but not the decrease phase, and temperature pat-terns cannot be correlated with any part of the cycle, weather is not the major factordetermining the regular population oscillations.3.3.3 Plant qualityMany plants have some form of response, chemical or otherwise, that is induced whenthey are attacked. When herbivores are at high population densities and the plantsare severely stressed by the herbivores' feeding, the plants can induce a response whichwill increase as the rate of feeding increases (Rhoades, 1983). In some cases, theseresponses can be detrimental to the herbivore and can cause reduced fecundity, death orgenerally affect herbivore population growth in some other manner. Thus the herbivorepopulation would decrease and, after some period of low herbivory, the plant qualitywould increase again, creating a delay in the system which could lead to herbivorepopulation cycles (Haukioja, 1980).Two models for plant-herbivore interactions in which plant quality is important21have been formulated. A theoretical model demonstrating that in plant-herbivore in-teractions the plant's quality could be a governing factor for cycles, was created byEdelstein-Keshet and R,ausher (1989). In the model, cycles occur for a limited set ofcircumstances. A detailed model incorporating food quality in the larch budmoth Zeiro-phera diniana system produced cycles that corresponded to the data for that system(Baltensweiler and Fischlin, 1988).However, the plant-quality hypothesis is probably not applicable to the tent cater-pillar system. In this system the quality of the trees that the caterpillars eat varies, butit is unlikely that it has major impact. Work by Myers (1981; 1988a) has shown that,even when plants are protected from severe feeding stress or are subject to simulateddrought conditions (both of which should cause the plants to have a higher quality),caterpillars do not survive better. If plant quality were the governing factor for cycles,the caterpillars should have a higher survival rate on plants with a higher quality.223.3.4 ParasitesMany models have explored the interactions between parasites and hosts. These in-clude the Nicholson-Bailey parasite model (see Section 2.5) and its many modifications.Anderson and May (1978; May and Anderson, 1978) created many models exploringdifferent relationships in host-parasite interactions. They found that the relationshipsthat produced cycles were a parasite-induced reduction in the host's reproduction ortime delays in the development or transmission of the parasite's infective stage (May andAnderson, 1978). Time-lags will usually generate cycles in a model (May, 1981), andare present in all systems since organisms cannot respond instantaneously to changesin nature. O3.3.5 Diseases: systems with immunityConsiderable theoretical work has been done on the effect of disease on animal andinsect populations. The models fall into two main categories, those in which the hostcan recover and sometimes become immune to the pathogen, and those in which thereis no recovery or immunity. The first type of model is used primarily for human diseasesbut could easily be adapted for any organism with an immune system. Bailey (1975)reviews many of these models, from the earliest epidemiology models of Kermack andMcKendrick to those published in the early 1970's. More recent work specializing innonlinear contact rates includes Hethcote and van den Driessche (1991), and Liu et al.,(1986; 1987). In general, these models assume a large, closed population, for whichbirth and death are ignored and most of the differences between models arises fromchanging conact rates or assumptions about immunity and recovery. There is also a23smaller class of models that incorporates a changing population size (Anderson, 1979;May, 1979). Both types of models (those with and without a varying population size)assume recovery or immunity. Some of these models have been reasonably successful atcapturing the cyclic dynamics of human measles and flu outbreaks (Bailey, 1975; Hasselland May, 1989).3.3.6 Diseases: systems with no immunityModels of systems without immunity from the disease are of particular interest whenstudying insects, since although populations of insects can increase their resistance to avirus (Briese and Podgwaite, 1985), the insects have no apparent immune system. Thesemodels assume open populations and include births and deaths. The disease modelsgive an indication of the many different situations in which the pathogen can regulatethe host, what factors are necessary for disease persistence, and when the system isunstable (Hassell and May, 1989; Anderson and May, 1981; Anderson and May, 1980,1981; review by Onstad and Carruthers, 1990). For cycles to exist, the equilibria must beunstable. Many disease-insect models have shown that imperfect vertical transmissionof disease will give population cycles of hosts (Anderson and May, 1980; Brown, 1987;Regniek, 1984; Diekmann and Kretzchmar, 1991; Hochberg, 1989). A more detaileddiscussion of some of these disease-insect models occurs in Chapter Disease and geneticsThe effect of disease on the genetics of a population is important in another hypothesis,formulated to explain the cycles in forest lepidoptera, and incorporating the variabilityin individual resistance to the virus. This hypothesis proposes a trade-off between twodifferent types of organisms, ones that have a high fecundity but are very susceptible tothe virus, and ones that are much more resistant to the virus but have a low fecundity.The susceptible individuals with high fecundity are selected for when the population isnear low density, so it is primarily these susceptibles that cause the population increase.However, when the population is high, more larvae become infected with virus and it isthe resistant type that survives the viral epizootic, and the population declines (Myers,1988b, 1990).This hypothesis involving the genetics of the organism is very similar to the ChittyHypothesis for small mammals, except that the Chitty Hypothesis proposes a trade-offbetween aggression and fertility: at high population densities, there are more organismsthat are highly aggressive but have low fecundity so the population declines and thoseanimals with high fecundity that are not as aggressive will be selected for and willeventually cause the population to increase again. This hypothesis appears to be likely tocreate population cycles. However, the Chitty Hypothesis has been extensively modelledand most models do not produce population cycles. Those that do generate oscillationsare often not biologically reasonable or only cycle for a very narrow range of parametervalues (Stenseth, 1981). One possible exception is the model by Hunt (1982, 1983). Sheshows mathematically that cycles are possible. However, it is difficult to interpret her25results biologically and to reproduce them through simulation. A model of geneticallycontrolled disease resistance and variation in fecundity for the tent caterpillars is similarto the models for the Chitty Hypothesis, and it does not produce cycles.264. INSECT-VIRUS MODELS4.1 IntroductionThere are many interactions that can give cyclic dynamics in models: predator-preyinteractions, weather-insect interactions, host-parasite interactions, plant-herbivore in-teractions and host-disease interactions. All the models referred to in the previouschapter, except the one by Wellington et al. (1975), are very general and do not nec-essarily consider the specific biology of any particular system. Indeed, as they havebeen formulated, they could be used to explain the population cycles either of insectsor of small mammals (except for differences in immunity to disease). Obviously, thebiologies of insects and mammals are different in many ways. It would therefore beinteresting to examine one hypothesis, or one set of models, in more detail and to applyit to a system and see if the cycles are robust. Here, disease models for insects will beconsidered. Four existing models will be examined and adapted to to details of tentcaterpillar populations.274.2 Anderson and MayAnderson and May (1980) modelled host-disease interactions using differential equationsfor continuous overlapping generations of classes of infective (I) and susceptible (S) in-sects, and a free-living infective stage of the pathogen (V). The model was formulatedin three equations. The first is for the susceptibles (S). The rate of change of the sus-ceptible population depends on the birthrate (a) of all individuals, both susceptible andinfected, minus the natural mortality (d) of the susceptibles minus the newly infectedindividuals.dS — a(S + I) — dS — 13V Sdt(4.1a)The term 13V S is the infection rate of a susceptible individual and is proportional to therate of encounters (3) between the susceptible and the free-living stage of the pathogen.The change in the infected population (I) results from the addition of the newly infectedindividuals minus dead infected individuals (natural mortality rate (b) + disease inducedmortality (a)). So,dIdt- = i3VS — (a + d)I. (4.1b)The third equation is for the free-living stages of the virus (V). These infective stagesare produced from infected hosts at a rate A and die at a rate p,. They are also lostthrough ingestion into susceptible and infected hosts. This gives,dV . AI — (p, + 0(S + I))V.dt (4.1c)Through analysis of this model the authors show that it is possible to produce limitcycle behaviour from the model. The crucial factor in the model is the mortality rate28of the free-living virus (u). If there were no free-living stage of the virus (so susceptibleinsects became infected directly from other infected insects), the model does not producepopulation cycles.The authors compared the results from this model to the behavior of the larch bud-moth, a lepidopteran that has cyclic dynamics. Given parameter values based on data,the model produced ten-year cycles, the same period as the larch budmoth populationshows. However, although the range in the predicted total population was similar to theactual population size (the log of the population size ranged from about 3.5 to 6.5 whilethe log of the actual population ranged from approximately 2 to 7), the predicted andactual prevalence of infection was quite different (between 30% and 50% in the actualpopulation compared with 80% in the model).Vezina and Peterman (1985) also used the Anderson and May model (equations 4.1)to model the Douglas-fir tussock moth ( Orgyia pseudotsugata), another lepidopteranspecies that has 7— 10 year population cycles. They simulated this model and severalvariations of the model that included more of the specific biology of the moth. Thefirst modification was to make the mortality rates density-dependent by changing theminimum natural death rate d to d' where d' = d c(S + I) (c is some measure ofthe severity of density-dependent natural mortality (Vezina and Peterman, 1985)). Thedisease-induced mortality a becomes a = m — d' where m is total host mortality. Thesecond modification was to add an incubation period between the time an insect becamediseased and when it infected others. A new class of organisms that are infected butnot yet infectious (E) was made. Organisms move into that class when they become29infected and move out at a rate v. The new system of equations is:dS a(S + I) – dS – 131/ S^ (4.2a)dtdEdt ^– (d v)E (4.2b)dI vE – (a + d)I^ (4.2c)dV = AI – (ft + 13(S + .0)V.^ (4.2d)dtThe third model Vezina and Peterman (1985) examined included vertical transmission.This means that not all the new larvae are susceptible: a proportion k of them are borninfected (and a proportion 1 – k of them are born susceptible). Thus, equations 4.1becomedS aS a(1 – k,)I – dS – Sdt—dI = pvs- (a ± d)1" akIdtdV =^– (,u, 13(S + I))V.dtFinally, they looked at a model in which all these components were combined.dt /.3V S – (d' v)E akEdIdt vE – (a + d')IAI – (ft + 13(S + .0)V.dtdS – aS all – k)./- – dS – /3V 5'dtdE(4.3a)(4.3b)(4.3c)(4.4a)(4.4b)(4.4c)(4.4d)Venzina and Peterman (1985) obtained realistic parameter values for the mod-els from field studies reported in the literature and adapted the values to fit into a30continuous-time model (so that all values were calculated per year and there would bediscrete generations). They found that it was impossible to obtain cyclic dynamics withappropriate periodicity when realistic parameters were used. They concluded that itwas not virus alone that generated population cycles so the models were not an adequaterepresentation of the tussock moth system. They hypothesized that a more successfulmodel would include the interactions between the tussuck moth and its enemies andfood.One problem with the models of Anderson and May (1980) and Venzina and Peter-man (1985) is that they assume that the transmission of the virus is proportional to theinitial densities of susceptible and infected individuals or to the densities of susceptiblesand free-living virus. This assumes that each instar is as infectious and resistant as allother instars. Dwyer (1991) performed some experiments in the field with the Douglas-fir tussock moth to determine the transmission coefficient (13) at different instars. Hefound that /3 was significantly higher for older instar larvae, indicating that transmis-sion over the entire year is not strictly proportional to the intial numbers of first instarlarvae. It is possible that if the models were adapted to account for a changing /3, cycleswould occur using realistic parameter values.314.3 BrownBrown (1987) created a general simulation model for insect-virus populations. Themodel was quite detailed and included a five day latent period (between becominginfected and dying and infecting others), an environmental pool of free-living stages ofthe virus, and a random temperature component that varied insect growth rates. Also,the amount of virus infected larvae produce when they die increased with the age ofthe larvae. In the model, the probability of becoming infected at any time depends onwhat Brown called the pathogen burden (`pb') and on the density of the hosts (`de').The pathogen burden is the number of infectious units that are available to infect hostsper host, or,pb th (4.5)where 'pa' is the number of available pathogens and `th' is the total number of hosts.Infection is considered to be a random process so a Poisson distribution is used, andthe equation for the probability of infection isp = 1 - e—de*pb (4.6)Notice that if the density (`de') is large, p will be close to one. Biologically, this simulatesgreater stress and increased susceptiblity at high densities (Brown, 1987). However, pis independent of the age of the larvae, and thus does not account for any increasingresistance to the disease.The model simulates overlapping generations (so that on any given day there areinsects of all age classes) and adult insects lay several eggs on each of several days.Both susceptible and exposed adults lay eggs, and they have the same fecundity. Even32though this simulation model shows oscillations, the cycles are unstable, with increasingamplitudes (Figure 4).I adapted this model to the tent caterpillars system by making the followingchanges.1. Exposed individuals have a lower fecundity than non-infected individuals. Theoriginal model assumed that exposed and susceptible adults would have the samefecundity, represented by the variable `ov'. I added a new variable `ove', to representthe fecundity of the exposed individuals, with `ove' < `ov'.2. The disease is carried by a proportion (q) of the eggs laid by the exposed adults(i.e., the larvae will be in the exposed class when they emerge). The rest of theeggs will be susceptible. Thus the equation governing the reproduction,S1 = ov * (s7 + E. )^ (4.7)becomesSegg OV * S7 + ove * Ea * (1 — q)^(4.8a)Eegg = ove * Ea * q.^ (4.8b)where:Si is the number of susceptible eggs in the original model,S7 , Ea are the susceptible and exposed adults,Segg Eegg are the susceptible and exposed eggs in the modified model.3. Reproduction only occurs towards the end of the season, with the adults laying allof their eggs in a single batch and then dying. The original model had the adults3311000Daysi i 10 500 1500Figure 4. Output from Brown's original model, showing unstable oscillations.34laying a few eggs each day of their adult life. Thus, any time an organism becomesan adult, she would lay a batch of eggs on that day and then die.4. Only the eggs and the free-living virus survive between seasons. I decided that themaximum total life expectency (from egg to adult) for a tent caterpillar was 100days. This means that at the end of the season, (at the end of day 100), all insectsin all classes except Segg and Eegg , are killed.This modified model no longer produced cycles. The simulated population stabilized orgrew exponentially, depending on the birth rates that were used and the reduction infecundity attributed to the exposed insects. If there only a small reduction in fecunditydue to infection, the population would be able to grow exponentially due to high fe-cundity. When there was a larger sublethal effect of virus on fecundity, the populationstabilized.354.4 Regniere4.4.1 Original modelR,egniere (1984) produced a model which most closely represents forest Lepidoptera.He used difference equations which included vertical transmission of the virus (k), norecovery from infection, and lower fecundity of diseased organisms due to sublethal ef-fects of the virus (Figure 5). Difference equations are more appropriate than differentialequations since the insects only reproduce once each year.In the model, the number of susceptible individuals at the end of the generationcan be calculated by examining three categories of organisms:1. The number of surviving susceptible individuals that did not contact the diseaseand gave birth to susceptible offspring ((1 — pt )cSt ). Since the function pi describesthe proportion of insects that become infected (see equation 4.10), the term 1 —p tgives the proportion of insects that do not become infected.2. Those susceptible organisms that contacted the disease but did not pass it onto their offspring ((1 — k)bpt St ). This implies that either these organisms werecarriers for the virus but they did not transmit it to their offspring, or that thesewere organisms that were more resistant to the disease.3. The surviving infected individuals that did not pass the infection to their young((1 — k)alt ).Adding items 1, 2, and 3 leads to the equation:St+i^(1 — k)(aIt + bpt St) + c(1 — pt)S t •^(4.9a)36f s a s (1-p)S (1-kieae p S (1-k)f.ai Iik^PSfeae kf. a. Ii^1disease free diseased Figure 5. A schematic diagram of Ileguiere's model (1984) (based on the diagram inhis paper). In the model, a b fe a,, and c Las . The top of the diagramrepresents generation t while the bottom represents generation t 1.37The parameters a, b and c are the net fecundities of infected, exposed and susceptibleindividuals respectively, and incorporate insect survival, fecundity, and the proportionof eggs that hatch. The net fecundities are related to each other by a < b < c, assum-ing that infected organisms are adversely affected by the disease more than exposed orsusceptible insects. Regniere assumed that a > 0, so that infected indivivals could re-produce. For the population to survive, c must be greater than 1. Vertical transmissionis included in the model in the parameter k, the proportion of offspring born to exposedor infected mothers, that are infected (k < 1).Similarly, the number of new infected individuals is calculated by the number ofsusceptibles that contacted the disease and gave birth to infected offspring (kbptSt),plus the number of infected individuals that also passed the infection to their offspring(ka/t ).= k(a/t bptSt) (4.9b)The proportion of insects becoming infected, Pt , is based on assumptions of randommovements of individuals and increases with the number of infecteds present.Pt = (1 — e—h-rt)# (4.10)where h is the rate at which the healthy larvae contact disease propagules (Regniere,1984). The parameter /3 determines the aggregation of disease propagules near thediseased insects, where 13 > 1. (Low /3 implies that the disease is highly concentratednear insects while high /3 indicates that the disease is more widely dispersed (Regniere,1984).) The notation is slightly changed from the original paper: Regniere used x andy where I have used I and S for clarity.38This model produces cycles with the eight to twelve year period similar to thoseobserved in natural populations. However, there are aspects of the biology of forestLepidoptera that are not yet incorporated. I modified the model to add some of theseaspects. In general, as the model became more detailed, it lost any tendency to cycle.4.4.2 ModificationsOrganisms that become infected at hatching probably die before reproducing (sinceinfected larvae usually live for about ten days (Entwistle, 1983a)). Therefore, I modifiedthe original model to include this fact:kbpt St (4.11a)St+i = (1 — Obpt St + c(1 — pt)S (4.11b)As shown in Figure 6, it was still possible to get ten to twelve year cycles in thepopulation, but with a narrow range of parameter values. Specifically, stable cycles(of any length) primarily occur only when /3 is close to one (the minimum value of 0as indicated by Regniere (1984)), meaning that the disease is highly concentrated nearinsects and the insects have a greater probability of contacting the disease. Biologicallythis would be the same as increaseing the amount of virus infected larvae producedwhen they died, or reducing the amount of virus needed for infection of a larva. When/3 is larger than one, (implying that the disease is more scattered), there is a muchnarrower range of parameters that will cause cycles. In many instances, the infectedorganisms quickly die out and only the susceptible ones are left. Because the diseaseis more widely dispersed, the chance of becoming infected is very low. Consequently,fewer infections occur and the virus disappears.39Figure 6 demonstrates the effect of changing three of the parameters. In all cases,c, the net fecundity of susceptible organisms, is held constant, and is equal to the uppervalue of b, the net fecundity of exposed individuals. If c were to change, the graphs wouldalso change, although the general pattern of the regions with cycles would remain thesame. Figure 6 also shows that it is not necessary to have any sublethal effect of thevirus (reduced survival or fecundity, characterised by b < c), for oscillations to exist.In Figure 6, a distinction is made between cycles and persistent oscillations (formathematical, not biological reasons). Cycles are those in which each oscillation isidentical with all others. Persistent oscillations are those in which the amplitude of eachcycle varies in a consistent, repeatable manner (for example, every third oscillation isidentical), and the periodicity is relatively consistant. These persistent oscillations arestable to small perturbations (i.e., if the population is perturbed, it will quickly returnto the same periodicity and amplitude as before the perturbation). Either of these typesof cycles could be relevant to natural populations, especially since in natural populationseach oscillation is not exactly identical to all the others.To get some idea of the effect of various parameters on the period and amplitudeof the cycles the model was simulated using one set of parameter values. Then oneof the parameter values was increased and the model was simulated again. Table 1shows a summary of the effects of increasing each of the parameters on the cycle periodand amplitude, and any constraints on the values that the parameters can take. Oneimportant constraint is the value of c the net fecundity of susceptibles, which must begreater than one for the population to survive. If c is less than one, the population is40unstable cyclesor no steady stateunstable cycles^r--^or no steady stateB)0.9k0.9kC)0.3^b 3Stable cyclesPersistent oscillations^ Stable equilibriumFigure 6. An example of the effects on the existence and period of cycles in themodified Regniere model (equations 4.11). as parameters k (vertical transmission), b(net fecundity of exposed individuals). and /1 (degree of clumping of disease) are varied.The numbers show the period of the cycles that are between eight and twelve years. Infigure A) # = 1,^B) = 2, and in C) = 3. In all cases, c 3.41not reproducing enough to replace itself, even if there was no virus present.Figure 7 gives an example of the oscillations that are seen before and after b, thenet fecundity of disease-exposed insects, is increased. When b 1, exposed organismsgive birth to exactly enough organisms to replace themselves since few of them survivebut those that do survive give birth to a large number of offspring (see Table 3). As canbe seen, the cycles become lower and shorter. This occurs because there are many moreinfected organisms being born, raising the infection rate for the susceptible organisms.Consequently, the total population will not become as dense as when this fecundityrate is lower. The cycles (as graphed on a log scale) look very different than naturalpopulation cycles, both in range and in pattern.Next, a free-living virus pool, V, was added to the modified model (equations 4.11).The virus has a survival rate, a, and w propagules are removed per susceptible as thesusceptibles become infected. Infected individuals contribute to this pool when they dieand release 7 virus propagules. Thus, the equation for the virus is:Vt+1 = av V + 7It — wPt St • (4.12)Since organisms are infected either from the free-living virus or from direct exposureto the dead infected individuals, the probability of contacting the infection should bechanged to reflect this.Pt = (1 — e— ht -h„Vt )0 (4.13)where hi and ht, replace the h of equation 4.10, and concern the rate at which a suscep-tible contacts the virus.42Parameter Meaning Period Amplitude Rangeb net fecundity — — < cc net fecundity — + > 1k vertical transmission + + < 1h contact rate = —a clumping degree + + > 1_av virus survival + + < 17 virus production + —w virus removal + +Table 1. Effect of increasing different parameters in the modified Regniere models(equations 4.11 and 4.12). A `+' indicates that the period length or amplitude heightincreased as the parameter increased, a `—' indicates that it decreased and an ' =' thatit did not change. The range indicates any constraints on the parameter, if any. Allparameters are > 0.430 10^20^30 40 50Effect of varying net fecundity of virus-exposed individualsGenerationsFigure 7. Effect of varying net fecundity of virus-exposed individuals on equations 4.11,the modified Regniere model. Increasing b gives lower amplitude, shorter period cycles.44The new model includes equations 4.11, 4.12, and 4.13. Cycles could occur buttheir periods and amplitudes were very sensitive to changes in some parameter values.As shown in Table 1, increasing the amount of virus produced by a dead infected insect(7) only decreased the height of the peaks. If the survival rate of the free-living virus(u,,) was decreased, cycles would appear in cases where there had previously been nocycles. As in the previous case (equations 4.11), varying b, the net fecundity of infectedinsects, changes the period and amplitude of the cycles. In addition, the existence ofcycles was very sensitive to the value of 0. Again, cycles with a 8-12 year period wouldonly appear for a narrow range of parameter values if was much different from one.Figure 8 shows the effect of varying k, b, and on cycle existence and period. Theparameters hi and hi, only affected the amplitudes of the cycles and not any of theobserved dynamicsIn terms of the biology of the system, these results indicate that if there is more viruspresent, the cycles should be shorter and the peak populations lower. This is becausesusceptible organisms would have a greater chance of becoming infected. Infected anddisease carrying organisms have lower survival and fecundity rates, so the populationsize starts to decline earlier (and does not grow as large). This argument fails only withvertical transmission (k). By increasing vertical transmission, the population is ableto reach higher levels at the peak of the cycle. Although this seems counter-intuitive,upon examination it can be seen that the low point of the cycle decreases as the verticaltransmission increases so the average population level decreases. Since the populationcannot increase as quickly, it take longer for the population to reach peak levels and45unstable cydesor no steady stateunstable cydesor no steady state0.3^b^3B)0.9k0.9kC)Stable cydesPersistent oscillationsStable equilibriumFigure 8. An example of the effects on the existence and period of cycles in the modelwhich includes a free-living stage of the virus (equations 4.11 and 4.12), as parametersk (vertical transmission), b (net fecundity of exposed individuals), and /3 (degree ofdumping of disease) are varied. Numbers show the period of the cycles that are betweeneight and twelve years. In figure A) = 1, in B) # = 2, and in C)/3 = 3. In all cases,c = 3.46the cycles become longer. The only exception occurs if transmission is perfect, in whichcase the population quickly dies out, since every insect that gets exposed to the virusgives it to her offspring, who will then die before reaching reproductive age.Even with changing parameter values in the model that included the free-livingstage of the virus (equations 4.11 and 4.12), the cycle periods were usually longer thanthose of most Lepidoptera (see Figure 8), and, as Figure 8 shows, the range of parametervalues that produced oscillations of the right period was narrow.Since equations 4.11 were the only modification that produced cycles of the rightlength for a fairly wide range of parameters, I concentrated on this version of theRegniere model. Regniere stated that his results were not dependent on the form of p t(the probability of becoming infected), although changing some parameter values suchas 0 changes other parameter values for the occurence of cycles. This was tested bycomparing three different equations for pt in which Pt depended only on the size ofthe infected population and not the size of the healthy population. The first p t is theoriginal one, seen in equation 4.10:Pt = (1^chit )13The second ishItPt = ( 1 +and the third is the negative binomial equation as described by May (1978):Pt = 1 — (1 + —hit) — '3(4.10)(4.14)(4.15)In all three equations, as the number of infectives becomes large, Pt —> 1, regardless of47the size of the healthly population. A graphical comparison of these models is shownin Figure 9 and a comparison of their results is in Table 2.Oscillations only occur in two of the cases although not necessarily for the sameranges of parameter values. For example, if equation 4.10 is used, stable cycles es-sentially only appear when /3 is close to one. When equation 4.15 is used, persistentoscillations occur if /3 > 2. For both of these equations, the value of /3 that producescycles in the model makes the value of Pt increase rapidly as the number of infectedindividuals increases. Generating cycles is more complicated when using equation 4.14.Although by varying the parameters h and /3 it was possible to get Pt to resemble thecurves from equations 4.10 and 4.15 (Figure 9), even after changing the other parame-ters the model did not produce cycles. So, the existence of cycles is dependent on howthe probability of becoming infected (P t ) is modelled.Further modifications of P t were done to include some dependence on the healthypopulation. Various forms were tried (Table 2). One of these assumes that the likelihoodof a susceptible organism contacting an infected larva is related to the total number ofencounters with all larvae (Nt ):hItPt 1^ + hNt(4.16)where N is the total population (S I). Thus, the probability of becoming infecteddepends on the proportion of infected individuals in the population. If there are very fewinfected organisms (compared with the total population), the probability of a susceptibleindividual encountering an infected one is small. Using this Pt , no cycles occurred; thesusceptible organisms always took over the population. This was because if the number48Pt Equation Cycles?(1 _ chit r 4.10 if /3 near 1i^hIt^V3 4.14 existence highly sensitive to parameter valuesk 1.-1-rt ,/1 — (1 + ilL) 4.15 oscillations if 0 > 2hIt 4.16 none1-EhN1^(1 +^—v St— vhst 4.17 narrow range of vTable 2. Summary of the various forms of Pt described in the text and results whensimulated in equations 4.11 (the modified Regniere equations).49O^I^ I10 20it InbledFigure 9. A comparison of the different equations for the probability of becominginfected that were used in the modified Regniere model (equations 4.11). In equation4.10, 0 = 1 and h = .5, in equation 4.14, 0 = 1 and h = 1 and in 4.15, 0 = 2.25 andh = .5. Cyles occur for equation 4.10, persistant oscillations in 4.15 and no cycles in4.14.50of infectives was low compared to the total number of hosts, the chances of contactingan infected individual would be small (Pt small). Few new infections would result,allowing the susceptible population to increase and become a larger portion of the totalpopulation. Eventually, the infection would disappear.The negative binomial form for pt that is used in equation 4.15 can also be modifiedto include some dependence on the number of susceptibles present. Hassell (1980) noteda relationship in his data between the number of hosts (susceptibles in the presentsituation) and the degree of dumping of the pest (virus here). He therefore decidedthat the dumping parameter (0) should not be a constant. The new form for p t thenbecame:hItPt = 1 — (1 +^ry s tSt (4.17)where v is the slope of the relationship between the degree of clumping and the numberof susceptible organisms. This is equivalent to stating that as the number of organismsincreases, they become more randomly distributed or that the infection becomes lessdumped. Cycles occurred when the model was run with this pt (equation 4.17) butthe cycles were highly dependent on the value of v that was used. If v was small (lessthan .5), the population stabilized. If a larger v was used, the population would showunstable cycles. There was a narrow range in between these two values that producedstable cycles.As can be seen, the manner in which the probablity of becoming infected is modelledhas a strong effect on the behaviour of the model. Biologically, this is a problem. Forthis model to be realistic, it is necessary to decide what form this probability would51have. Ideally, this could be determined from data of population sizes and infectionrates. However, upon examination of Figure 9, it is easy to see that each of the threefunctions would fit data equally well (assuming first that these data could be collected),and that it would be virtually impossible to determine which, if any, was the trueprobability function.Two hypothesis can be made about the model. First the model is a close rep-resentation of the tent caterpillar system. In this case, each of the functions for theprobability of infection could be tried in the model in turn, and the one that caused themodel to produce cycles similar to those that are seen in nature would be the one thatrepresents what is actually happening. The second, more likely, hypothesis that couldbe proposed is that since the model is so sensitive to how the probability of becominginfected is modelled, the model must not be an accurate representation of the importantparts of the Lepidopteran system.I originally ran the models with arbitrary parameter values, usually based on pa-rameter values used in Regniere's paper. To see if the model could produce cycles usingrealistic parameter values, the parameter values were changed to reflect actual data fromtent caterpillars more closely. This was done by simply choosing constant parametervalues that seemed to be within the ranges of the means for different years. For exam-ple, average fecundity ranged from 140 to 220 eggs per batch, the early survival rate(eggs to fourth instar) varied between .2 and .6, and the late survival rate (fourth instarto adult) ranged from .005 to .2. I assumed that the susceptible larvae that had beenexposed to the disease would have lower survival than those that had not been exposed52to the disease and if they survived would have lower fecundity. Thus the parameters band c, the net fecundities (survival x fecundity), could be estimated from the data bychoosing lower fecundity and survival rates for the exposed insects than for the suscep-tible insects. Table 3 lists parameter estimates for fecundity and survival. Parameterb, the net fecundity of exposed insects was calculated by multiplying together all thevalues in the column marked 'low' while c, net fecundity of susceptible organsims wascalculated by multiplying the columns labelled 'high'.Table 3 only reflects how two of the parameters were chosen. Most of the otherparameters were the same for all simulations. The parameter h, the rate at whichhealthy larvae contact disease propagules, has no effect on the period (Table 1) so itsvalue is inconsequential. I chose /3 = 1 since from previous simulations, that was thevalue that resulted in the highest incidence of cycles (for example, see Figures 6 and8). For each combination of b and c, I allowed k, the vertical transmission rate to varybetween 0 and .9 (i.e., for each set of b and c, I simulated the model ten times, once foreach k). The parameter values that I used, and the results from the model are shownin Table 3.When using parameter values that were within the ranges of the means mentionedabove, cycles occurred in only one instance and for only one level of vertical transmission(k). I also chose to use some fecundity values that were lower than any of the means.This seemed reasonable since a low mean value allows for the possibility of some of theindividual fecundities being quite low. Cycles occurred in two of the three instancesmentioned in Table 3. In one case, the cycles persisted for a wide range of vertical53Fecundity Early Surv. Late^Surv. Parameter Cycles?low high low high low high b c 8-12 yr50 200 .3 .5 .2 .3 3 30 no50 200 .3 .5 .04 .2 .6 20 k = .3-.9*100 200 .3 .5 .04 .2 1.2 20 k = .9*120 210 .3 .5 .01 .01 .36 1.05 no140 220 .34 .46 .04 .2 1.9 10 k = .8*140 250 .2 .5 .01 .02 .28 2.5 no150 200 .3 .6 .2 .3 9 36 no* (8 or 9 yr cycles)Table 3. Parameter estimations for b and c (the net fecundity of exposed and susceptibleindividuals) based on data, and the results from various simulations of model 4.11 usingb, c as given in the table, /3 = 1 and k varying from 0 to .9 (i.e., each set of b and c,was simulated ten times, once for each k).54transmissions levels. The fact that oscillations occurred more frequently when a verylow fecundity level is assumed indicates that if the cycles are a reasonable representationof the natural cycles, the virus must have a large impact on the fecundity of the exposedorganisms.Although cycles can be generated with realistic parameter values, there are twofactors about the cycles that make it unlikely that the model is realistically capturingthe dynamics seen in the field. The first factor is that the cycles in the modelled popu-lation had a very different form from that seen in natural populations. Real populationoscillations tend to show a few years at the peak and a few years near the bottom of thecycle (see Figure 14) rather than the sharp peaks and valleys that are demonstrated inFigure 7. Most insect-virus models show cycles similar to those in Figure 7.The second point is that cycles were not the general case when using relativelyrealistic parameter values (or even unrealistic ones) and the existence of the cycles issensitive to the parameters that are chosen. Table 3 shows that changing the survivalrate at only one stage can alter whether or not cycles are present. For cycles to beconsidered to be possible in the field, they must be robust to a range of parameterssince in the real world, the parameter values differ each year. If virus is important ingoverning the cycles, there must be some important part of the virus-insect interactionthat is missing from this model.55KI4.5 Diekmann and KretzchmarRecently, Diekmann and Kretzchmar (1991) produced a continuous time disease modelthat is based on two classes of organisms (not necessarily insects), susceptibles (S) andinfectives (I), and incorporates various aspects of birth, death and infections. Althoughcontinuous time models are not usually applicable to univoltine organisms such as in-sects, I adapted the model to fit some of the characteristics of the lepidopteran system.The infection rate 0 is a function which depends on the proportion of infectives inthe total population.(4.18)where IC represents the contact rate between susceptibles and infectives. There are twotypes of mortality: natural mortality (it) and mortality caused by being infected (a). Anunusual aspect of the Diekmann and Kretzchmar model was that the cyclic dynamicsdepended on the way in which reproduction was modelled. The authors assumed thatboth male and female infected adults would contribute to some reduction in fecundity,and that this effect was multiplicative (i.e., if a susceptible and an infective adult mated,the reduction in fertility would be 4" while if two infecteds mated, this reduction wouldbe e) . This leads to the system of equations:dS  0 S2 + 2 ,5/ + e /2dt^S + I^µS — OSdITit- = os - (a + p)I(4.19a)(4.19b)The authors show that the behaviour of the model can be summarized using sixdifferent phase-plane diagrams (Diekmann and Kretzchmar, 1991). Which diagram is56applicable to any case depends on the parameter values that are used, especially theparameter K, (contact rate). The authors give complicated formulas for determining,without simulation, what the behaviour of the model will be based on the parameters,and they describe six possible behaviours. Only two of these six cases contain thepossibility of cycles. In both cases, the value of 4. is small, less than .5 (meaning thatthe disease has a large effect on the fecundity of infected organisms). One of the cases,the cycles are independent of initial conditions and may be biologically relevant forsome system. In the second case in which cycles occur, the behavior is dependent onthe initial conditions as well as the parameter ranges. This second case has no clearbiological interpretation.Contrary to the model, in the forest Lepidoptera only the females contribute to anyreduction in fecundity if infected. However, the existence of cycles is entirely dependenton the form of the birth term (Diekmann and Kretzschmar, 1991). When pairwiseformation is not included, or when only the infected mother causes reduction in fertility,the termfrom equation 4.20a becomesS2 + 2V - S + 4-2 120 (S + .0 (4.20)8(S + 4-I). (4.21)It can be shown that when equation 4.21 is substituted into equation 4.19a, only two ofthe six behaviours described by Diekmann and Kretzchmar (1991) are possible, neitherof which include cycles. In one case, the population stabilizes at an equilibrium whilein the other, the population grows exponentially although the proportion of infected57individuals in the population stays the same. From various simulations using differentforms for birth, I found that in general, if the birth term is written as 13Sf(x) wherex a necessary condition for cycles to occur is f(x) < 1. This does not meanmuch biologically but if equation 4.22 is rewritten in this form asbirth = ps f (x)^ (4.22)with^f (x) = 1 4.x^ (4.23)it is easy to see that for all positive values of 4, f(x) > 1 in equation 4.23.Removal of pair formation eliminates the major cause of reduced population at highlevels of infection. When the number of infectives is high, the birthrate under the pairformation model (using equation 4.20) is quite low while in a modified version, equation4.21 for example, the birthrate is relatively high. This is demonstrated in Figure 10a.To compensate for this, the infection term (0) can be altered from its form in equation4.18 to:c+ s2 +12 (4.24)in an effort to reduce the population at high levels of infection. A visual comparison ofthe two forms of 0 can be seen in Figure 10b. As 0, the rate of infection, increases, moresusceptibles will become infected thus increasing the death rate, reducing fecundity,and reducing the population level. Even these modifications did not cause the modelto produce cycles without pair formation.5820^40^60^80^100# Infected0^5^10^15^20# InfectedFigure 10. (a) A comparison between the birth term which assumes that both infectedmales and females contribute to the reduction in fecundity (equation 4.20) and thebirth term in which only infected females affect reproductive output (equation 4.21).The second equation increases with the number of infectives much more rapidly thanthe first. In the graph, e = .3. (b) A comparison of the different forms for the infectionrate ek that was used in the model after the birth term was modified. Note that equation4.24 gives a much higher infection rate than the original, equation 4.18.595. A NEW MODEL5.1 IntroductionModels of disease dynamics in the literature either are not appropriate to virus-lepid-opteran interactions, or oscillations no longer occur when the models are modified toresemble the biology of forest Lepidoptera more closely. Therefore, I developed a newmodel which was based on the biology of the tent caterpillar. If the model were toproduce reasonable results with one lepidopteran species, it could be adapted for otherspecies. The formulation of this model is based in part on the idea of Hochberg etal., (1990) to separate the within-season dynamics (infections) from the between-seasondynamics (reproduction). This leads to one of the most important differences betweenthis model and any of the models that were discussed in Chapter 4 (except Brown's,(1984)) which is that the probability of infection can vary during the larval growingseason as the number of larvae, the susceptibility of the larvae, and the amount of viruschanges.This chapter describes the new model. The first section derives the model and givesthe basic equations that are used. The second section discusses how the basic parametervalues were derived. Results from the simulations and the analysis are in the followingchapter.605.2 Formulation of the model5.2.1 Model overviewThe tent caterpillar-virus model is concerned with the dynamics of individuals withindifferent family groups (tents). There are three types of larvae: susceptible (S), exposed(E), and infected (I). Susceptible larvae have not come into any contact with thedisease, while exposed larvae have actually consumed a full dose of the virus but havenot become infected. The exposed larvae could also be called carriers since even thoughthey do not die from the virus, they can pass it on to their offspring. In the model,there is no immigration or emigration of caterpillars or of virus. In nature, infectioncan occur in two ways: horizontally or vertically. In the model, horizontal transmissiontakes place in the larval stage as the infection is passed from one larva to another.Vertical transmission occurs from mother to offspring and is represented in the modelduring reproduction. The model is divided into three periods: larval, pupal and adult.Schematically, the overall model is shown in Figure 11. All parameters and variablesused in the model are also described in Table 4.61Adults^VirusVa" vdeathB)1,4V, death/- a e .deathP41- as deathA) reproduce,dieEggsoverwintereLarvae 3,E1(Instars 1-5)Pupae(1-a p )deathFigure 11. The lifecycle of the tent caterpillars with the corresponding parameters andvariables from the model (see Table 4). A) Overall lifecycle. B) Larval stage.62Variable Meaning UnitsSEIVTHsusceptibleexposedinfectedvirustentstotal hosts (S + E + I)larvae/tentlarvae/tentlarvae/tentpropagules/tentnumberlarvae/tentParameter Meaning Units Where Usedt instar (mini-intervals) time 5.1y generation time 5.8-10as survival rate of susceptibles 5.1a-,crysurvival rate of exposedssurvival rate of virus5.15.1a-P survival rate of pupae 5.6Pt probablility of exposure to virus 5.30 degree of dumping of virus 5.3q proportion of exposures resultingin infection5.1wt virus ingested propagules/larva 5.4'Yt virus produced propagules/larva 5.5a scaling factor 5.5dy density-dependence 5.7a threshold at which density-dependence becomes important5.7.f.9 eggs hatched to susceptibles #/susceptible 5.9fe eggs hatched to exposeds #/exposed 5.10k vertical transmission 5.10c proportion of new tents inareas with no virus5.11Table 4. Variables and parameters used in the model.635.2.2 Larval StageThe first stage is the larval stage. Since there are five instars, this stage is dividedinto five mini-time-steps, designated by the subscript t. The interactions between thevirus and the caterpillars occur within each such interval, and are pictured in Figure11b. Since every interaction in the larval stage takes place within a year, the year (orgeneration) subscript (y) does not change. Each interaction also takes place within atent or a type of tent (described in Section 5.2.4).Within the course of each instar, the larvae have probability P t of coming in contactwith the virus, and thus becoming exposed or infected (a proportion q of larvae thatcontact virus become infected, the rest (1 — q) become exposed), or a chance 1 — Ptof avoiding the virus and remaining susceptible. The larvae will survive from othermortality sources at a rate u, or (re , depending on whether the larvae were consideredsusceptible or exposed at beginning of the instar. Thus, the number of susceptibles atthe end of an interval is determined by the proportion of susceptibles that survive theinstar (us ) and that did not come into contact with any virus (1 — pt ).Sy,t+1 (1 P-t)Crs Sy,t (5.1a)The probability of infection, Pt that is used here depends on the total population andon the amount of virus present, and is explicitly defined later.The assumptions are similar for the exposed larvae. Of the survivors, some will nothave contacted any virus (1 —pot ), or will have contacted virus but not become infected(p t (1 — q)). In either case, they will remain in the exposed class. There are also some64surviving susceptible larvae that contacted virus but did not become infected.E y ,t+1 = pt (1 — q)( 0" s Sy ,t Ue Ey ,t) + (1 — Pt)C e Ey ,t (5.1b)The number of infected larvae is based on the number of surviving susceptible andexposed larvae that become infected after contacting some virus. The infected larvaeall die during the time interval. Thus, the equation for the infecteds is:ly , t+1 = pt g(u8 Sy3t 0-eEy,t) (5.1c)Virus is removed from the environment by organisms when they contact the virus(since infections and exposures result from ingesting the virus) and through death.Infected larvae release virus into the environment when they die. Consequently, theamount of virus at the end of a period is determined by the surviving virus (Cry Vy,t )5minus the virus that was ingested by susceptible and exposed larvae (w t ), plus the virusthat is released by the newly infected organisms when they die (-y t ).Vy,t-1-1 = Civ Vy,t 'Yt ,t+1 WEPt(0 Sy,t Cre Ey ,t) (5.1 d)The amounts of virus ingested or released by larvae vary with the instar. Their formsare given in equations 5.4 (wt ) and 5.5 (7t ). Since all infected larvae die within the instarperiod, and since the amount of virus depends on the new number of infecteds (4, t+1 ),equation 5.1c could be eliminated and substituted into the virus equation (equation5.1d).The term Pt , the probability of exposure to the virus, which governs the transmis-sion of the virus between organisms within a tent (horizontal transmission) varies with65time. The expression for pt is based on the distribution of the virus. Suppose thereis Vy , t amount of virus present near a tent containing Hy ,t larvae. Of this virus, thereare Vy , t /w t doses (i.e., there is enough virus to infect at most Vy , t /w t larvae). Assumethese doses are distributed near the larvae according to the negative binomial distribu-tion with a mean Vy , t /(w i liy , t ) and clumping parameter 0. A low value of /3 impliesthat the virus is highly dumped near the larvae, while if p is large, the virus is morerandomly distributed. In practice, it is very difficult to estimate 0. The probability ofthe larvae escaping all the virus is given by:Po = (1 + VY ' t ) -13 .13Wt Hy ,tTherefore, the probability of a larva encountering the virus at any time is:Pt = 1 - (1 +ii„, VYT_T' t )-#pw-t y ,twhere Hy , t = Sy ,t + Ey,t + 4, t , the total population.There are other functional relationships that vary with time. The amount of virusthat is ingested by an organism (wt ) increases with time. This is equivalent to increasinga larva's resistance to the virus as the larva grows. The amount of virus produced bya larva when it dies ('yt ) also increases with the size of the larva at death. The actualequations for these relationships is not important to the dynamics as long as both areincreasing and the amount of virus produced by a larva is greater than the amountingested (ryt > wt ). After discovering this, I used the forms:lot = 10t -4^(5.4)7t = awt (5.5)(5.2)(5.3)66where t is the number of the instar and ranges from 0 to 4 and a is some number greaterthan one, used to ensure that -y t > wt . The derivation of equations 5.4 and 5.5 andtheir relationship to data is discussed further in Section 5.3, equations 5.16- Pupal StageThe second part of the model involves the pupae. I assumed that any infected larvaedie and do not pupate, while the exposed and susceptible larvae pupate. Pupal survival(rnp ) was the same for exposed and susceptible organisms.Sy ,6 = Crp Sy ,5 (5.6a)Ey,6 = CfpEy , 5 (5.6b)4,6 = 0 (5.6c)On the right hand side of the equations t = 5, representing fifth instar organisms (therewere five intervals in the first part of the model). Adults, denoted by the subscript 6,emerge from the pupae. This part of the model is not essential to the general dynamics,since it only scales the numbers of susceptible and exposed larvae downwards and re-moves any remaining infected larvae. Therefore, this segment could easily be combinedwith the third section of the model.675.2.4 Adult StageThe final stage of the model is the adult stage, in which reproduction and overwinteringoccurs. Overwintering is only implied in the model since the winter mortality of eggsis included in the birth terms, f8 and f6 . Although it is unrealistic, I assume that thevirus has no mortality during the winter. The consequence of changing this assumptionis discussed in the next chapter. The number of surviving adults is found for each type:susceptible and exposed. I assume that half of the adults are male and do not producenew tents. This assumption does not affect the overall dynamics. At this stage thereis some density-dependence (dy ) which is based on the total number of tents from theprevious year as follows:dy a + Ty • (5.7)The term a determines the level at which the effects of density-dependent becomeimportant. The term dy will be small if there is a large number of tents and consequently,it accounts for possible effects from crowding such as increased mortality or inability ofthe females to find viable oviposition sites. This term will affect the number of adultsthat will lay eggs and hence, the number of new tents.Since a tent is formed by the offspring of a single female, the number of tents eachyear is directly related to the number of females that lay eggs which hatch. The numberof tents can thus be determined by adding the number of adult susceptible and exposedindividuals (denoted by the subscript 6) in all tents, dividing by two for the number ofadult females, and accounting for density-dependent effects using dy :Ty+1 =2—1 (SY ' 6 + Ey,6).^ (5.8)68For most simulations of the model two types of tents are used: those that containonly susceptible larvae and those that contain both susceptible and infected individuals.Female susceptibles produce egg masses which, when they hatch, will contain onlysusceptible larvae. These tents will be in areas where there is virus already. Thenumber of eggs that a female susceptible lays that hatch is denoted by the parameterfs •Type 1 (susceptible, virus present):dSy+1 ,0 =2(5.9a)Vy+1,0^Vy ,6^ (5.9b)Adults exposed as larvae produce offspring that are a combination of susceptibleand infected, the proportion determined by a parameter k (vertical transmission). Iassumed that all these egg masses were laid in areas that already contained some virus.Exposed adults lay fewer eggs than susceptible adults due to sublethal effects of thevirus. This gives the second type of tent:Type 2 (combination, virus present):Sy+1 , 0 = -2-(1 — k)fe Ey , 62dyly +1 , 0^ICJ e-C-fy,62Vy+1,0 Vy,6 + 74+1,0(5.10a)(5.10b)(5.10c)where f e is the number of eggs that an exposed female lays that hatch. The valueson the left-hand side of equations 5.9 and 5.10 are the number of larvae in the nextgeneration in each of the tent types.69In determining both types of tents, I assumed that they are in areas that alreadycontain some virus. At low population densities that may be an erroneous assumption.Consequently, in a few simulations, I also examined the behaviour of a model in whichI had added two other types of tents: ones that contain only susceptibles and ones thatcontain both susceptibles and infecteds. Neither of these types of tent were in areasthat already had virus present. In determining all four sets of tents, I decided that aproportion c of each type of tent would be laid in areas with no virus. The new set ofequations are:Type 1 (susceptible, virus present):sy+ 1,0 = 2 (1 — cVsSy , 6^ (5.11a)Type 2 (combination, virus present):—dy(1 — c)(1 — k)fe Ey , 62cA(1 — c)k fe .Ey , 62Sy-f-1,0-4+1,0Type 3 (susceptible, no virus present):S +1 0 = --v-cfs Sy,6Y^2(5.11b)(5. 1 1c)(5.11d)Type 4 (combination, no virus present):Sy +1,0 = 441 — k) feEy ,6^ (5.11e)-4+1,0 =  2 ck fe Ey , 6 (5.11f )The presence of tent type 4 (tents with both susceptible and infected larvae butwith no virus present) did not affect the dynamics, and only marginally affected the level70at which the population stabilized. However, tent type 3 (tents containing susceptiblesonly with no virus present) did affect the dynamics. Type 3 tents, since there was noinfection present, either in the larvae or in the environment, acted as a refuge from thevirus, causing a higher population level and more stable dynamics. Consequently, mostsimulations were done assuming only two types of tents (equations 5.9 and 5.10). Ifcycles were to occur, they would be most likely to arise with only the two tent typespresent.715.3 Parameter valuesMost of the parameter values that were used in the simulation were originally basedon values mentioned in the literature, mainly from Entwistle (1983a) and Myers (pers.comm.). The parameter values that are given below are a basic value used in mostsimulations, and in all cases, simulations were also done with different values (See Section6.2, Table 5).The survival rates of the larvae at various stages are loosely based on data fromMyers (pers. comm.). In the data, the averages of the early survival rates of thelarvae (up to the fourth instar) range from .12/season to .82/season. Since some of themortality is due to natural causes and some to infection, the values for a-, and Te, thesurvival rates of the susceptible and exposed larvae respectively, were chosen from thehigher end of the scale and were calculated by choosing values for early survival ratesof susceptible and exposed larvae as .7/season and .5/season respectively. These valuesare then raised to the power 1/5 to get the survival rate per instar, assuming that thesurvival rate is the same at each stage. Thus,a, = .7 1 / 5 = .93/instar (5.12a)cre = .5 1 /5 = .87/instar. (5.12b)The survival rate of the virus can be calculated either from its half-life or from theproportion of virus remaining after a certain length of time. Entwistle (1983a) statesthat NPV virus has a half-life of one day but that there is more than 1% of free-livingvirus left after twenty days. For both these statements to be true, the death rate of thevirus cannot be a strict exponential decrease. In fact, virus mortality usually depends72on how exposed to the sun it is. However for convenience, exponential decay will beassumed. Given the statements above, two possibilities arise. Using the hypothesis thatafter twenty days there is is at least 1% of the virus left, the survival rate of the virus,a, during one instar (ten days) is:o-, > .1/ten day period (5.13a)If the half-life of the virus is one day, the survival rate becomes:cry = .001/period. (5.13b)In most simulations, a value of .01 was used, although the effects of the higher and lowera, values were also explored.In the model, I assumed that the pupal survival rate was the same for exposed andsusceptible organisms. From the data, late survival rates (which includes pupal andadult) range from .001/season to .2/season. For most simulations, the pupal survivalrate was taken to beo-1, = .1/season. (5.14)At the time of reproduction, I assumed that a proportion k of the offspring fromexposed adults are infected at birth. The value of the parameter k cannot be determinedfrom the data so a value of .7 was used for most simulations (i.e., 70% of the offspringfrom exposed organisms are infected at birth). The number of eggs laid per female isgiven in the data with a range of 140 to 230 eggs/batch. I assumed that the exposedadults would lay fewer eggs than the susceptible adults (140 vs 230), and that only 60%of the eggs from either type of adult would hatch. The parameters f8 and f, reflect the73number of eggs laid and hatched by susceptible and exposed females respectively, andfor most simulations were determined to be:^f, = 140^ (5.15a)fe = 100 (5.15b)As mentioned before, the amount of virus ingested and released by a larva varieswith the age and size of the larva. Although experiments for determining the amountof virus needed to infect a larva 50% of the time (LD 50 value) are not always reliable,they give some indication about the levels of virus that may be important. Entwistle(1983a) gives a range of lepidopteran LD 50 values as from 10 propagules when the larvaefirst hatch to 10 6 propagules for fifth instar larvae. (A propagule is a virus polyhedralinclusion body (PIB) which is the virus surrounded by protein and is visable with amicroscope (Entwistle,1983a).) Assuming that the infection rate increases by an orderof magnitude at each stage, the relationship for the amount of virus ingested per larva(million propagules/larvae) as given in equation 5.4 iswt 10t -4 (5.16)where t indexess the instar and ranges from 0 to 4. Since w t is the amount of virus thatwill cause infection 50% of the time, the proportion of contacts resulting in infection(q) was taken to be 50% (or q = .5).Entwistle (1983a) also gives information for the the amount of virus per milligramthat lepidopteran larvae can produce and for the range of total virus per larvae that canbe produced. Since the model does not calculate the weights of the larvae in the tents,74the range was initially used as the basis for determining how much virus was producedby the infected larvae. Entwistle (1983a) gives the range as 10 9 to 1.5 x 10 10 . I assumedthat the lower end of this range corresponded to the larvae that died while in the firstinstar and that the upper value was the amount produced by the fifth instar larvae. Ialso assumed that the progression from one to the other was linear. Thus, measuringthe amount of virus produced by an infected larvae in millions of propagules per larvae,7t = (1 + 3.5t) x 10 3 (5.17)where t is a measure of age and ranges from 0 to 4. When the model was simulatedusing this relationship for the amount of virus produced, the population usually died outbecause of these high production levels and because I assumed that all virus that wasreleased, and that did not die, was available to the larvae. To alleviate this difficulty,I changed the production curve to make it a function of the amount of virus ingested.Thus, as given in equation 5.5,7t = aw t (5.18)where a = 4.5 (a value determined through simulation). This makes the lower end ofthe range for the virus produced much lower than given by Entwistle (1983a).Since there were no data for the degree of clumping of the virus (0), the levelat which density becomes important (a), and the proportion of organisms born in anenvironment with no free-living virus present (c), the parameter values used were chosenfor other reasons. The parameter /3 must be greater than 0 and I assumed that it mustalso be relatively small (compared with oo) so I chose 3 = 1 for most simulations. Theproportion of larvae in areas with no free-living virus is also likely to be small so I used75c = .2 for the simulations in which more than two tent types were possible. I let a = 500since for most simulations, it caused the population to be near levels that were close torecorded population sizes.766. SIMULATION RESULTS AND ANALYSIS6.1 IntroductionTo determine the possible behaviours of the model, I simulated the model on the com-puter and analyzed a simplified form of it. Although the analysis of the simpler modeldid not indicate that cycles were possible, oscillations occurred in some simulationsof the full model. However, these oscillations had different characteristics than areobserved in natural populations.In the first section, the results from various simulations are examined. The secondsection lists the important results from the analysis of a simplified version of the modeland their interpretations and implications in biological terms. The actual analysis isshown in Appendix 2. The applicability of the analysis to the actual model is alsodiscussed. In all cases, as a first step in deciding whether or not the model 'worked',cycles, preferably ones with an eight to twelve year period, were the criterion. If thiscriterion were fulfilled, I then examined the predicted and the average clutch sizesthat the females laid and related them to the ranges given in the data. Finally, I alsocompared some of the patterns in the changes in fecundity and survival that arise duringthe model's cycles with those in actual cycles.776.2 Simulation ResultsSimulations were done with the basic parameter values derived in Section 5.3 and byvarying the parameter values individually as shown in Table 5. In all cases the popula-tion stabilized at some level, either at zero (no organisms present) or at a higher level.This population size was always stable; if the population was perturbed away from thatlevel, it returned rapidly. Table 5 shows the effect of varying each parameter individu-ally on the level at which the total population H or the amount of virus V stabilized.Persistant oscillations did not occur under any circumstance that was examined whenonly one parameter was varied within the ranges indicated in Table 5 and all othervalues were at the base levels shown in the table.Some of the behaviours given in Table 5 may seem counter- intuitive. One such caseis the fact that when the virus survival rate (up) is increased (implying that there shouldbe more virus present), the amount of virus present when the population stabilizes actu-ally decreases. This occurs because the caterpillar population has also declined. Thus,fewer larvae become infected, and they become infected at a younger age. Consequently,not as much virus is produced since in the model, the only way the amount of virus canincrease is by being eaten by a larvae which dies. A similar argument can be made forthe effect of the vertical transmission rate (k).The behaviour of the model was very sensitive to some of the parameters that wereused in the model. As shown in Table 5, these were: q, the proportion of exposures tovirus that result in infection and a, a parameter used in calculating the amount of virusthat is released by an infected larvae when it dies. The sensitivity was such that if the78Parameter BasicValueRangeExploredStableHLevelVSensitivity CyclesPeriod Amplitudeas .93 .75 — .99 + + s +a-, .87 .75 — .99 — + s + —vv .01 .001 — .2 — — + +o-P .1 .01 —.2 + + s — +k .7 0 — 1 + +f8 140 100 — 400 + + s +fe 100 50 — 200 + + — —q .5 .3 — .9 — — v + —a 4.5 1-10 — — va 500 50 — 1100 + + = +13 1 a — 10 — — + +s 15-40% change in population level with 10% change in base parameter value.v 60-90% change in population level with 10% change in base parameter value.Table 5. Effect of increasing different parameters on the level at which the populationstabilizes and on the period and amplitude of cycles in the new model. The basic valuewas determined in Section 5.3 and was used for all simulations except where noted. Therange column indicates the range through which the parameters were varied. A `+'indicates that response increased, a `—' that it decreased, and an `=', that it remainedthe same. Base parameter values used to calculate the cycles were: = 4, f, = 320,a-, = .1 and all others as in this table.79values were much higher than were given here, the population would die out and if theywere much lower, the infected part of the population would disappear. Thus, all resultsare highly dependent on these parameters.The model was also simulated by varying more than one parameter at a time fromthe base levels. In some cases, eight to twelve year cycles did occur (Figure 12). Incases when oscillations occurred, I examined the effect of varying different parametersto determine their effect on the period and amplitude of the cycles (Table 5). Figure 12gives an example of the effects of changing some of the key parameters: ay , the survivalrate of the virus, /3, the degree of dumping of the virus, fs and fe , the net fecundities ofthe susceptible and exposed adults. The period of the cycles, once they had occurred,was not very sensitive to changes in the parameter values (Figure 12). In general, thecycles increased in length as a, and were increased and decreased slightly as f3 andfe were increased. Increasing the susceptibles' birth rate (18 ) allowed the population torecover more quickly from the decline since the susceptibles were the largest proportionof the population when the population was at low levels.The cycles occurred when 13, 18 , and cr, were much higher than the base valuesgiven in Table 5. However, this does not necessarily indicate that the parameter valuesthat produce oscillations are unrealistic. For example, the parameter 13, the degree ofclumping of the virus near the insects, cannot be determined from the available data andwould be difficult to determine empirically. I had originally assumed that since duringthe season the virus is found primarily near tents (since the larvae do not wander farfrom the tent), that /3 would be small, close to one. The values of /3 that are shown in80ma_v ..05, fe=100 fe 50 s ma v6^12^12 11.5^11^11^115^12 11.5^11^11^11^114^12^11^11^11^11^113^11^11 10.5^10.5^10^102 10^10 10 9.5B)65beta 4321160^200sigma_v . .1, fe240^280fs100320 360 160 200 240 280 320 36010101010^1010^9.59.5^9.59.5^^200 240^280 320 360fsC) sigma_v . .2, fe . 100 D)65beta 4321fe 150, sigma_v65beta 4321 160 200 240 280 320 360fs160 200 240 280 320 360fs7 8-12 year oscillations>12 year oscillations decaying oscillationsFigure 12. Occurrence and period of cycles in the model as the parameters /3 (degreeof dumping of the virus), o -, (virus survival rate), f, and fe (number of larvae bornper susceptible and exposed females respectively) are varied. Cycle lengths have beenaveraged so a period of 9.5 indicates that the cycles alternated between 9 years and 10years in length. A * indicates that the cycles were extremely irregular but tended toremain between eight and twelve years.81Figure 12 are of the same order of magnitude as one and therefore may still qualify as`small' (especially since the maximum value of /3 is oo).As discussed above, based on information in the literature av , the survival rateof the virus, might be around .001/period or .1/period. Although the base value wastaken to be a, = .01/period, the higher values of o-, that produce cycles are closer tothe upper estimate (e.g., o-t, = .05, .1 or .2). If u„ < .01 (closer to the lower estimate),cycles did not occur. The upper calculated value for a, was based on the observationthat there was still more than 1% of the virus left after a certain amount of time. Ifthe simulated cycles are a reasonable approximation of natural oscillations, it impliesthat observations about the presence or absence of virus may be more reliable thancalculations of half-life. Actual virus likely does not show exponential decay since thelifetime of the free-living virus is usually determined by the amount of exposure to sunand rain it receives. If the virus is protected, it will live longer than if it is exposed.In the model, I assumed that the virus had no mortality during the winter. Themodel was very sensitive to this assumption. If I allowed less than 90% of the virusto survive the winter, the cycles disappeared, unless the value for the within seasonsurvival rate, cry , was increased. However, increasing a„ to a level at which cycles wouldagain occur, usually leads to a„ values that are much larger than data suggest.Finally, the values for f,, the number of new susceptibles per susceptible female,were also much higher than originally estimated. To see if these values could be rea-sonable, the maximum and minimum average number of eggs per female was calculatedduring the course of a cycle for the instances in which Figure 12 shows that cycles82occurred (Figure 13). These values were then compared with the range shown in thedata. In most cases, the average clutch size was much larger than the data indicated,especially under the assumption that not all the eggs hatched. In fact, almost all theinstances in which the dutch size was similar to the data occurred if egg survival wasperfect (i.e., if all the eggs that were laid, hatched). One exception to this occurs ifthe number of new larvae from an exposed female (f e ) is very small (Figure 13E). Innatural populations, the egg hatch rate is near 100% (Myers, pers. comm.). Therefore,the birth parameters that produce oscillations are reasonable.Next, the cycles were examined to see if they were similar to natural oscillationsin more ways than the period and the range of dutch sizes. It is not necessary toexamine the predicted number of tents or population size since this can be easily adjustedby altering a, the parameter that determines the population size at which density-dependence starts to have an effect. This parameter does not affect the period or averageclutch sizes of the cycles. Instead, the patterns of population increase and decrease wereexamined. In natural populations, it has been observed that the average dutch size doesnot start to decrease until the first year of the population decline (Figure 14A,C). Thisis not observed in the simulated cycles (Figure 14B,D). The cycles produced by themodel show clearly that the average clutch size starts to decrease before the population,and that it is this reduced clutch size that causes the population decline.I then compared the early survival rates (from egg to fourth instar) of the naturalpopulations and the simulated population since the populations in all cases declined inthe same generation but the declines were not all due to reduced fecundity. The survival83240 280 320 360fsegg survival rate = 1C) sigma_v = .1, fe = 1006 1751302901503301552101402051452001555beta43170130170140290150325155285155275165325160315170200 240 280 320 360C)6F)65beta45beta43 32egg survival rate = 1sigma_v = .2, fe = 100290105335105290 335110 105240 285 330115 110 110200 240 280 320120 125 120 115190 270 310130 130 135 135240 280 320 360Isegg survival rate = 1fe = 150, sigma v = .1205185245195280200315205240 275 315195 205 210235 270 310205 210 215255 295230 235240 280 320 360Isegg survival rate = 1A) sigma_v = .05, fe = 100205 240 280 320165 175 180 185235 270 310185 195 200290230fsegg survival rate = 1fe = 50, sigma_v = .1145701857522575265803058034580145 185 225 265 305 34575 75 80 85 90 95145 185 225 265 305 34575 80 95 90 90 95140 180 220 260 300 34080 85 90 95 100 105195 270 310120 135 140160 200 240 280 320 360fs245120310125375130445130510135580135240 310 375 440 510 575120 130 135 140 145 160•^240 'i . 305 370 440 505 570125 ::. 135 140 150 155 160716729 365 43----36406g-145 155 160 170 175320 385 450 515200 210 225 235D)65beta4325beta43265beta4egg survival rate = .6E)^fe = 50, sigma v = .16160 200 240 280 320 360fsFigure 13. Maximum and minimum clutch sizes produced during each of the cyclesindicated in Figure 12. In A, B, C. D, F, egg survival is 100%. In E, egg survivalis 60%. Areas shaded grey roughly correspond with the actual data range of 140-230eggs /clutch.84-2A1^I 1 1 1 1 1111111111^11^2 3 4 5 6 7^5^9^10^11^12^13^14^IS^16^17Model0NqN00CM0• B\ \^1 .....^\\ \^n^..; \^\._.// .^... ,- .•# tents# eggs/batch# tents# eggs/batch# tents# eggs/batchWesthamC0 -g111111111111111^2^3^4^$^6^7^6^9 10 11 12 13 14Model-CC.)CCS.0Cl)C)# tents# eggs/batchi^.. •. •. .. .^•. • ..^. .. . .. . .. .^.. . .• . .• , .•- 2- 81^I^1^1^1^I^I^I^I^1^1^I^1^I1^23^4^56^7^59^10 11 12 13 1400a)0Mandarte11111111111111111123455 7 5 9 10 11 12 13 14 15 16 17GenerationsFigure 14. Log number of tents and average clutch size in two actual populations: A)Mandarte, C) Westham and a simulated population (B.D). Parameter values used inthe simulation were: /3 = 4, f, = 280, fe = 100, a = 200 and all others at the basevalue given in Table 5.85rates in the natural population declined before those of the simulated population (Figure15), and it is likely that these lower survival rates contribute to the natural populationdecline. Therefore, although the population levels are cycling with the same period, theoscillations produced by the model are a different type of cycle than is seen in nature.Some of the assumptions that were used in deriving the basic parameter values maynot be valid. Therefore, some of them were changed slightly to see the effect on thebehaviour of the model. The first of the changes was made to the survival rates of thelarvae. The basic parameter value given in Table 5 is a constant, assuming that eachinstar has the same mortality rate (due to all causes other than virus) as each of theothers. If instead it is assumed that the survival rate is lower for the younger instarsand higher for the older instars, and if the overall survival rate is the same (.7 or .5),the dynamics are not greatly affected.I also assumed in that the amount of virus that larvae ingested (that gave them a50% chance of becoming infected) was a function that increased linearly on a log scale.Various other functions for the amount of virus ingested were also tried, including somethat decreased or remained constant as the larvae grew. Some of these are pictured inFigure 16. As long as wt was increasing, the model produced approximately the samebehaviour, and this was the only instance in which cycles occurred. This is support forthe knowledge that larvae need more virus to become infected as they grow larger.Originally the model was simulated by differentiating between each individual tent.Each tent initially either contained only susceptible organisms or contained a mixture ofsusceptible and infected organisms and each tent was in an area with differing amounts86ci)ca)rn0-JO(V) ••■• • /^•\^ •• ••B•••••^•^•• --•-Mandarte0riOOOOOOO\1.^.. .^.. ....... ■■•^ ^.^ :^./..^.; 0. ..^.^):• •. .^:^'....^.'.. : .. ....^/^..'. : ^•- cc! ‘t.-1 2 3 4 5 6 7 8 9^11^13^15 17Modelco1 2 3 4 5 6 7 8 9^11^13^15^17GenerationsFigure 15. Early survival rates (from egg to fourth instar) in a simulated and a naturalpopulation. Generations are the same as in Figure 14A and B. Parameter values usedin the simulation are indicated in Figure 14.87..-..^ /.... ..,..• /. ....... a^-/. --N..^. ...^ //N .. / CN^.//^ /N.^. ^./ /N.)•:7..,^./^\^ / , /, d//^ N. z// •../ / N% / N./ z/N../^ N.N/ ...,^ N./^ .// .-,--- N ./ ....°^ ..N,.•-•--, N....-0.•••• N.2^ 3^ 4^ 5InstarFigure 16. A comparison between the different forms for w t , the amount of virusingested at each instar. Lines a, b and c give similar results in the model (includingcycles). Line d does not give cycles. Line e usually allows the population to die out.88of free-living virus. The model was also simulated by only keeping track of the numberof organisms in each of four different groups of tents (see equations 5.11) or in each oftwo distinct types (see equations 5.9 and 5.10). Finally, the model was also simulatedby ignoring all distinctions between tents.The division of the model into individual tents was not critical to the behaviourof the model although the separation into different tent types was important. Themodel behaved in the same way using individual tents and using four different tenttypes. However, when the model was simulated with no separations between tents, itbehaved quite differently; the dynamics were much less stable, showing more incidencesof decaying cycles, and a greater likelihood of the population becoming extinct. Ignoringthe differences between tents means that the population is modelled as if there areinteractions between all the larvae and as if virus can move between sites. When thepopulation is divided into different tent types, there is still the implication that virus canmove between tents that are of the same type, but that there is no interaction betweenlarvae from different types of tents. Those populations that start with both virus andinfected larvae present will have more virus present, fewer larvae and therefore higherprobabilities of larvae contacting the virus than tents of a different type.The only difference between dividing the population into two types of tents andfour types of tents was that, as mentioned in section 5.2, using four groups of tents gavea more stable, higher level population. For example, all the cycles shown in Figure 12were generated from the model that assumes that all eggs are laid in areas in whichthere is virus already present. If I assume that some of the tents are in areas in which89there is no virus previously (using four types of tents), cycles do not occur. There aretwo exceptions to this. If the proportion of tents laid in areas with no virus is very small(c < .1) or if the only eggs that are laid in areas with no virus contain infected larvae,then some cases of shorter period, smaller amplitude cycles do occur. The differencesbetween using two or four types of tents occurs because one of the categories of tentscontains only susceptible organisms with no free- living virus nearby. This class ofdisease-free tents acts as a refuge from the virus. The organisms in these tents haverelatively high survival and birth rates, since there is only natural mortality and nodisease-induced reduction in fertility. Therefore, when this class is present, it has astabilizing influence on the overall dynamics6.3 Analysis6.3.1 IntroductionThis section examines the possible behaviours of the model determined by analysisperformed on simplified equations. The details are given in Appendix 2. The firstsection discusses the reproductive rate of the virus, and its effects on model dynamics.Next, I calculate the conditions for stability of the simplified model, and discuss theimplications for the more complex model.906.3.2 Virus reproductive rateFirst, I calculated an approximate reproductive rate (R t ) for the virus. This is therate at which the virus will increase in one within-year time-step. The calculations areshown in Appendix 2.2, and they give the result:Vy ,t+1 = RtVy ,tRt = 0"v (7tq - wt)(0- .9Sy ,t + Cr e E y ,t) +whereH y ,t = Sy ,t + Ey , t + 4, 2 .It is well known that, for simple difference equations such as the logistic, the re-productive rate, r, of an organism must be at least one for the organism to survive,and that increasing r will lead to cycles (May, 1976; see also Figure 1). However inthis model, Rt varies with time so the product of the Rt 's must be at least one (i.e.r = Ro R1 R2 R3 R4 R5 > 1). This means that any single R t may be much less than one.Since the calculations were done for very low levels of virus, r is only accurate as anapproxiation to the actual reproductive rate of the virus if V is small, less than 10 -4 .Since virus is measured in millions of propagules (i. e., if V = 10 - 4 then there are 100propagules present), this is not an unreasonable requirement.wtHy,t(6.1)(6.2)(6.3)916.3.3 StabilityIn order to determine analytically the possible behaviours of the model, the model hadto be simplified. Since pt , the probability of contacting the virus, was the most complexterm, it was simplified by setting p = a constant. The stability conditions could theneasily be determined for a modified version of the model in which p was constant andin which all tent types were combined into a single type. The results of the analysis ofthe simplified version will not give the exact behaviour of the current model, but it maygive some clues as to possible behaviours or key parameters.From Appendix 2.3, the stability condition is given in equations A2.14 and can besummarized to say that for the population to stabilize at some non-zero level, ib > 2where7,b fa x + (1 — ic)fez (6.4)where x and z are combinations of parameters only (see equations A2.19 and A2.22).This can be interpreted easily in terms of the biology of the system. The term lkrepresents the reproduction of the organisms. In formulating the model, I assumed thatonly half the organisms give birth. Therefore, there must be at least two offspring permother for the population to continue. The conditions state that if there are fewer thantwo offspring per mother, then the population will die out (go to the trivial equilibrium),and if there are more than two, the population will go to the upper equilibrium. If Ihad assumed a different sex ratio in the model, the stability conditions would reflectthe new sex ratio. Basically, for the population to survive, the females must give birthto enough offspring to replace themselves and those adults that do not give birth.92The upper equilibrium is determined by d, the term governing the density-depend-ence in the model. If there is no density-dependence acting at the time of reproduction(i.e., d = 1 always), there is only one equilibrium value, the trivial one, where thepopulation size is zero. This means that the population would either become extinct(if the trivial equilibrium were stable) or would grow without bound (if the equilibriumwere unstable). Clearly, due to the way that the model is formulated, the density-de-pendence is necessary to stabilize the population.Simplifying the model by allowing p to be constant leads to the unreasonable as-sumption that the amount of virus present at any time does not contribute directly tothe stability or instability of the system since the modification causes S and I to beindependent of the virus (see system A2.10). However, even with the simplification, theconstant value for p that is chosen can reflect the amount of virus present, so a small pimplies that there is little virus present, and the population will stabilize at a relativelyhigh level, while if p is dose to one, there is a large amount of virus present, and thepopulation will either become extinct or stabilize at a low level. The relationship be-tween p and the level at which the population stabilizes is shown in Figure 17. Thus,the value of p that is chosen (and indirectly the amount of virus) affects the size of thepopulation at equilibrium, and even whether or not the population will survive.In summary, the analysis of a simplified version of the model indicates that cyclescannot occur. The analysis shows that the population will die out if the females do notlay enough viable eggs to equal the total adult population. If there are more than thisnumber of offspring produced, the population will grow to some level determined by the930.8 1 .00.0^0.2^0.4^0.6Infection probability (p)Figure 17. The relationship between the constant value of p (the probability of becomingexposed to the virus) and the number of susceptibles at equilibrium, according to theanalysis. The point pc is where the equilibrium population size is zero.94effects of density-dependent reproduction. The results from the analysis do not accountfor all the behaviours that are observed in model simulations so it can be assumed thatthe fact that Pt , the probability of becoming infected, varies with time is an importantcomponent in the observed dynamics.6.3.4 Comparison with results of simulationsIn the first part of the analysis, the virus reproductive rate r was calculated and it washypothesized that increasing r would lead to cycles. In fact, with all parameters at thebase values given in Table 5, increasing r just lowers the level at which the populationstabilizes until eventually, for high enough r, the population becomes extinct. Thisoccurs because as r increases, there is more virus present, allowing more organisms tobecome exposed or infected and either to die or to have a reduced fecundity. Thus thepopulation size will be lower. However, as observed in Figure 12, if other parametersare also moved away from the calculated base value, increasing r leads to cycles, andhigher r produces longer length cycles.There are two weaknesses with the analysis. The first is that all tent types arecombined into one type. As seen from simulations, the behaviour of the combinedmodel is different from the behaviour of the separated model. The model was lessstable using a single tent type. The second, more important defect of the analysis isthat it assumes that p, the probability of exposure to the infection, remains constantregardless of how much virus is present.In spite of these inadequacies, the results can be extended to some degree to theactual model where p t varies during the season, and between years (equations 5.1 and955.3). From many simulations, it was determined that the geometric mean of the pt foreach time i within one season, will give a reasonable estimate about whether or notthe population will stabilize (or die out), as long as the population is near equilibrium.For example, given a set of parameter values, the critical value pc can be estimated.The value pc is the p for which the non-trivial equilibrium equals the trivial equilibrium(i.e., the population dies out). This point is indicated in Figure 17. When the modelis simulated, the geometric mean, 15 of the pt values for each step can be calculated asp = (ponp2p3p4) 1 / 5 • If p is less than pc , then the population will stabilize, otherwiseit will die out. Thus, the analysis can be applied (if the population is already nearequilibrium) by using p as the constant p.Cycles are observed from the calculation of p by examining its value over time. Ifthe population is going to stabilize, p will be fairly similar in each time step. If cyclesoccur, then p will be less than pc for several time steps (allowing the population tomove towards the steady state) and then will be greater than pc for several steps (asthe population declines).Not all the results from the analysis carry over to the actual model. If the popu-lation stabilizes, it has fewer organisms present than were predicted from the analysis.This occurs for two reasons. The first is that po and pi , the pt values for the first twotime-steps of the season, are often elevated due to vertical transmission of the virus andbecause the young instar larvae are more susceptible to the virus. These Pt values causehigh levels of early mortality to a large number of organisms. The second reason isthat the analysis was done assuming all tents had the same amount of accessible virus.96When the model is simulated, some tents have a large amount of virus present whileothers have little or none. This causes high variation in mortality between tent types,and usually results in having fewer organisms present than predicted from the analysis.6.4 SummaryFor this new model, given in equations 5.1 to 5.10, eight to twelve year cycles canoccur under certain parameter combinations (as in Figure 12). In most cases, if Iassumed that not all eggs hatched, the average clutch sizes were much larger than isseen in natural populations. With the assumption that all eggs hatched, there were moreoccurrences of clutch sizes with a reasonable range (see Figure 13). However, the rangeof parameters producing cycles of the right length with reasonable average clutch sizeswas very narrow. Also, the cycles showed a different pattern than is seen in naturalpopulations (Figures 14, 15). Simulated cycles showed that the population declinedafter the average clutch size decreased while the reverse is true in natural populations.Also, in natural populations the survival rates dropped earlier in the cycle than inthe simulated cycles. However, from the simulations and analysis, several features ofstructure and values of parameters are important in creating cycles.1. Density-dependence affecting the proportion of adults reproducing.Without any sort of density-dependence in the model, the population did not cycleor stabilize at any level. Birth rates are high enough (when using parameters basedon data as in Table 5) so that even with virus reducing fecundity, the populationis able to grow quickly and the susceptibles become the major proportion of thepopulation.972. Sub-lethal virus affecting fecundity levels.If the virus does not affect fecundity, then f8 = fe . However, cycles never appearunder this circumstance, even if there is a large sublethal effect on survival in thelarval stage. It is the lowered fecundity that is the primary cause of the populationdecline in the model. In fact, the larger the difference between f8 and fe , themore likely that cycles will occur (Figure 12). It is not necessary to also have thesublethal effects on survival during the larval stage in order to produce cycles.3. Survival rate of free-living virus.If there is not enough virus present in the environment (because the survival rate istoo low or because it does not survive over the winter), it allows the population toescape and grow to some stable level. Conversely, if the survival rate of the virusis too high, the population could start to show erractic cycles or become extinct.The parameter k controlling vertical transmission has a similar effect in that as itincreases, more virus will move into the next generation and keep the populationlower.4. Infection rate.The fact that the analysis and the results disagree implies that the fact that theprobability of infection varies with time and density is crucial to the productionof cycles. Another critical factor included in the infection rate is the degree ofclumping of the virus near the larvae, something that is difficult to measure.987. CONCLUSIONSModels were used to test the hypothesis that the eight to ten year population cyclesthat are found in tent caterpillars could be produced by a virus. Four insect-virusmodels from the literature were examined. Each of these models initially showed thatpopulation cycles were possible. All of these models lost their tendency to cycle ifrealistic parameter values were used. The same results were found by Vezina and Peter-man (1985) who adapted to Anderson and May (1980) model for biologically resonabledetails of the douglas-fir tussock moth.Since none of the existing models produced reasonable cycles, I created a newvirus-insect model with particular attention to the tent caterpillar system. This modelincluded many characteristics about the biology of tent caterpillars and the interactionsbetween the larvae and the virus. One of these characteristics not included in anyprevious model examined was that the caterpillars' susceptibility to the virus changesas the larvae become older. Thus, the probability of becoming infected varies duringthe larval growing season, depending on how much virus is available and the age of thelarvae. Heterogeneity was incorporated in my model by varying the amounts of virusavailable to caterpillars.The model was examined both through simulations and through analysis of a sim-plified version of the model in which the probability of becoming infected was heldconstant. This analysis showed that no cycles were possible in the simplified version ofthe model. However, simulations of the model showed that cycles could occur. Thisimplies that is important that the probability of becoming infected (p) varies with the99amount of virus and number of larvae present.Cycles occurred when using relatively realistic parameter values and the periodof the cycles produced by the model corresponded well with those of actual cycles.However, the simulated oscillations showed that the average clutch size started to declinebefore the population size decreased and survival rate decreased at the same time as thepopulation. In natural populations, average clutch size does not go down until the firstyear of the population decline while the survival rate decreases before the populationdecline. Thus there is a difference of one to two generations between the simulated andactual cycles of dutch size and survival rates. There are three explanations for this:1. There are aspects of the biology of the virus or the virus-insect interaction thathave not yet been discovered but that are crucial to the character of the cycles.2. There is some factor or factors, not including virus, that is or are governing orinfluencing the cycles.3. Virus works with some other component to produce the patterns seen in the cycles.Each of these arguments will be examined in turn. First, there may be some crucial,as yet unknown biology about the virus or the manner in which the virus interacts withthe insects. This conclusion affects how the model is actually constructed, so under thisconclusion, it is still possible that virus is the single cause of population cycles and thatthe only reason that proper oscillations have not yet been produced is that somethingis missing or inaccurate.This unknown biological factor could involve either interactions that are occurringor the measurement of key parameters. In the first case, it is possible that the virus is100transported between tents in some manner (by parasites or wind, for example) or thatthe virus has some other detrimental effect on exposed larvae besides reducing adultfecundity. Also, little is known about the free-living stage of the virus. As mentionedpreviously, there is some debate as to how long the virus can live in the environment,and whether the free-living form is available to first instar larvae. Finally, some of themechanisms in the model, such as the probability of becoming infected, may change asthe number of organisms increases. For example, as the population size grows, the larvaewill be more likely to encounter larvae or virus from another tent and so increase theirlikelihood of becoming infected. The probability of becoming infected would also changeif the susceptibility of the larvae changed during the cycle. Research in these areas mayproduce different or additional results than are portrayed in the current model.The second area included in the first conclusion is the measurement of variousparameter values that seem to govern the interactions. Many aspects of the biology ofthe virus are difficult to observe or to measure. For example, often it is not possible totell if caterpillars have been infected by the virus until after they have died. Therefore,it is difficult to measure transmission and infection rates. As a consequence, the effectof virus on insects that have been exposed to the disease but have not died from it, isnot well understood. Since there is always variability among individuals in the field,if different larval mortality rates or adult fecundities are measured, it is difficult todetermine why these differences arose, or how much of the difference may be attributedto virus (e.g., are two values different due to differences in environment or due to asublethal effect of virus?). The survival rate of the virus and the degree of clumping of101the free-living virus near larvae are both critical parameters in the production of cycles,but are almost impossible to determine. These are all aspects of the biology that can bestudied further and it is possible that when searching for clarity in the determination ofparameter values, additional information about interactions between the virus and thelarvae will be discovered.The second possible explanation states that there may be some other factor orfactors, aside from virus, that is or are acting to create the cycles. Many of these otherpossible components, such as plant quality, weather, genetics, or parasites, have beenmodelled before by other researchers and are discussed in Section 4.2. None of thehypotheses seem entirely plausible for the tent caterpillar case unless one of the factorsbehaves similarly to the virus in its effect on the larvae (such as reducing fecundity).For example, if a mother experienced bad growing conditions, possibly from weather orfood, she may have reduced fecundity or her offspring may have reduced survival. It ismore probable that it is one or more elements acting together to produce cycles. Forexample (although this has not been modelled), it may be that weather and parasitescombine to produce oscillations. Few models that have been produced examine theeffect of multiple factors, excluding virus, on insect populations.The third explanation is essentially a sub case of the second since it suggests that itmay be virus acting with at least one other system component to create the cycles. Thesevarious combinations could include possibilities such as weather and virus, parasites andvirus, plant quality and virus or even weather, plant quality and virus. It also may bethat at high densities the larvae become much more susceptible to a variety of mortality102factors, including disease and that such mortality factors would act to decrease survivalrates as are seen in the natural populations.Several models have examined the possibility of a multiple component interactionproducing cycles (Berrryman, pers. comm.; Hochberg et al., 1990; Wellington, et al.,1975). Berryman maintains that the cycles are caused by the action of the parasitesand that the virus just increases the magnitude of the cycles. Hochberg et al. (1990),show that cycles are possible when both a parasite and a pathogen are present in thesystem. As discussed in Section 3.3.2, Wellington et al. (1975) produced a modelthat combined effects of parasites, virus and weather. That model could be furthertested using more recently collected data. These models provide initial support for thehypothesis that more than one factor creates the cycle. Further modifications of themodels are necessary and some support from a natural system is still needed to see ifthe models are valid for the tent caterpillar system.The results from my model also support this third explanation. Using only virus,it was possible to produce cycles with the right period. The model cycles differed fromnatural cycles in that the population decline was caused by a virus build-up which led tomore exposed individuals and lower average fecundities, leading to a population decline.In the natural system, the average fecundities do not decrease until after the populationdeclines and it is lowered survival rates that seem to influence the population decline.Therefore, if my model is a reasonable representation, some other factor (not includedin the model) may be working with the virus to reduce population levels before thevirus can reduce the average fecundity. In particular, some component in the natural103system must be acting to reduce survival without affecting fecundity.The new model, like many others, is a purely deterministic model, and assumesthat all parameter values such as survival rates and basic fecundities remain constantfrom year to year. In nature, this is obviously not the case. There are always stochasticevents occurring which may increase or decrease any of these values, or change some ofthe relationships. A valid deterministic model should not be sensitive to small changesin parameter values. If the stochastically induced changes are small, many of the resultsthat have been discussed remain similar but instead of all the results being smooth, thereis variation around the basic lines. However if these changes are large, the deterministicmodel no longer produces similar results and is not an adequate representation of thereal world. Finally, the use of deterministic models makes it easy to explore varioushypotheses and to see the effect of various changes or assumptions. It would be possibleto add to and to adapt this model to further examine some of the above explanations.There are always more modifications that can be made to any model and morepossibilities that can be explored. One obvious change that could be made to my modelis to alter the clumping parameter /3. It is likely that in a natural population, does notremain constant. At low populations, virus will be confined to the areas near the tentsand would be low. However, at high populations, tents and caterpillars are abundantand there are more interactions between caterpillars of different tents. The virus willthen be less likely to be as clumped near particular tents, and the parameter may behigher.Another modification to the model would be to add some other factor that affects104the tent caterpillars, such as parasites, to see if the addition of this factor would causethe population to cycle in the same patterns as are seen nature. Ideally, this factorwould act to reduce survival levels, but not fecundity, near the peak of the cycle. It islikely that simply adding further detail to the virus-insect interactions would not havethis effect. Finally, since this model is developed specifically for tent caterpillars, itwould be interesting to adapt it for other Lepidoptera to see if any of the results couldbe generalized.The results from the model also suggest key parameters that ideally should bemeasured in the field. Two of these are related to the free-living virus. If the incidenceof free-living virus (i.e., where it is found) could be measured, the clumping parametercould be estimated (see Krebs, 1989 pp81-89). Another key parameter was thesurvival rate of the free-living virus, both during the larval growing season and duringthe winter. This parameter is also key in Hochberg et al.'s (1990) model for insect-pathogen interactions. Other parameters that it would be useful to measure would bethe vertical transmission rate (what proportion of a mother's offspring are infected),and the fecundity of females exposed as larvae.1057. LITERATURE CITEDAnderson, R.M. 1979. The influence of parasitic infection on the dynamics of host popu-lation growth. in Population Dynamics, 20th Symposium. Brit. Ecol. Soc. London,1978 (eds Anderson, Turner, and Taylor), Blackwell.. 1991. Discussion: The Kermack-McKendrick epidemic threshold theorem. Bull.Math. Bio. 53:3-32.Anderson, R.M. and R.M. May. 1978. Regulation and stability of host-parasite popula-tion interactions. I. Regulatory processes. J. Anim. Ecol. 47:219-247.. 1979. Population biology of infectious diseases. Part I. Nature. 280:361-367.^. 1980. Infectious diseases and population cycles of forest insects. Science 210:658-661.. 1981. The population dynamics of microparasites and their invertebrate hosts.Philos. Trans. R. Soc. London. Ser. B. 29:451-524.Andrewartha, H.G. and L.C. Birch. 1954. The Distribution and Abundance of Animals.University of Chicago Press, Chicago.Bailey, N.T.J. 1975. The Mathematical Theory of Infectious Diseases. 2nd ed. MacmillanPub. Co., Inc. New York.Baltensweiler, W and A. Fischlin. 1988. The larch budmoth in the Alps. in Dynamicsof Forest Insect Populations. (ed. A. Berryman) Plenum. Pub. Corp. pp 331-351.Briese, D.T. and J.D. Podgwaite. 1985. Development of viral resistance in insect popu-lations. in Viral Insecticides for Biological Control. (ed. K. Maramorosch and K.E.Sherman) Academic Press, Orlando. pp 361-98.Brown, G.C. 1987. Modeling. in Epizootiology of Insect Diseases. (eds J.R. Fuxa and Y.Tanada) John Wiley & Sons. pp 43-68.Clark, E.C. 1955. Observations on the ecology of a polyhedrosis of the Great Basin tentcaterpillar Malacosoma fragilis. Ecology 36:373-376.^. 1956. Survival and transmission of a virus causing polyhedrosis in Malacosomafragile. Ecology 37:728-732.^. 1958. Ecology of the polyhedrosis of tent caterpillars. Ecology 39:132-139.Daniel, C. 1990. Climate and outbreaks of the forest tent caterpillar in Ontario. M.Sc.thesis, University of British Columbia.Diekmann, 0. and M. Kretzschmar. 1991. Patterns in the effects of infectious diseaseson population growth. J. Math. Bio. 29:539-570.Dwyer, G. 1991. The roles of density, stage, and patchiness in the tranrnission of aninsect virus. Ecology 72:559-574.Edelstein-Keshet, L. 1986. Mathematical Biology. Random House Inc. New York.Edelstein-Keshet, L. and M.D. Rausher. 1989. The effects of inducible plant defenses onherbivore populations: 1. Mobile herbivores in continuous time. Am. Nat. 133:787-809.106Entwistle, P.F. 1983a. Control of insects by virus diseases. Biocontrol News and Infor-mation. 4:202-228. Commonwealth Institute of Biological Control.^ . 1983b Viruses for insect pest control. Span. 26.Finerty, J.P. 1980. The Population Ecology of Cycles in Small Mammals. Yale UniversityPress, New Haven.Hassell, M.P. 1980. Foraging strategies, population models and biological control: a casestudy. J. Anim. Ecol. 49:603-628.Hassell, M.P. and R.M. May. 1989. The population biology of host-parasite, host para-sitoid associations. in Perspectives in Ecological Theory. Princeton University Press,Princeton. pp 319-347.Haukioja, E. 1980. On the role of plant defenses in the fluctuation of herbivore popula-tions. Oikos 35:202-213Hethcote, H.W. and P. van den Driessche. 1991. Some epidemiological models withnonlinear incidence. J. Math. Bio. 29:271-287Hochberg, M.E. 1989. The potential role of pathogens in biological control. Nature337:262-265.Hochberg, M.E., M.P. Hassell and R.M. May. 1990. The dynamics of host-parasitoid-pathogen interactions. Am. Nat. 135:74-94.Hunt, F. 1982. Regulation of population cycles by genetic feedback: existence of periodicsolutions of a mathematical model. J. Math. Bio. 13:271-282.. 1983. A mathematical analysis of the Chitty Hypothesis. in Population Biology.(eds H.I. Freedman and C. Strobeck) Lecture Notes in Biomathematics. Springer-Verlag. 52:33-40.Hutchinson, G.E. 1978. An Introduction to Population Ecology. Yale University Press,New Haven.Kermack, W.O. and A.G. McKendrick. 1927. Contributions to the mathematical theoryof epidemics-I. Proc. Roy. Soc. 115A:700-721.Kingsland, S.E. 1985. Modelling Nature. University of Chicago Press, Chicago.Krebs, C.J. 1989. Ecological Modelling. Harper & Row, New York. pp 81-89.Liu, W.M., H.W. Hethcote and S.A. Levin. 1987. Dynamical behavior of epidemiologicalmodels with nonlinear incidence rates. J. Math. Bio. 25:359-380.Liu, W.M., S.A. Levin and Y. Iwasa. 1986. Influence of nonlinear incidence rates uponthe behavior of SIRS epidemiological models. J. Math. Bio. 23:187-204.Malthus, T.R. 1798. An Essay on the Principle of Population. in An Essay on thePrinciple of Population (ed A. Flew) Penguin, England, 1970.Martinat, P.J. 1987. The role of climatic variation and weather in forest insect outbreaks.in Insect Outbreaks. (eds P.Barbosa and J.C. Schultz) Academic Press, Inc., SanDiegoMay, R.M. 1973. Stability and Complexity in Model Ecosystems. Princeton UniversityPress, Princeton.107^. 1978. Host-parasitoid systems in patchy environments: a phenomenological model.J. Anim. Ecol. 47:833-843.^. 1981. Models for two interacting populations. in Theoretical Ecology 2nd ed (ed.R.M. May) Blackwell Scientific Pub. Oxford.May, R.M. and R.M. Anderson. 1978. Regulation and stability of host-parasite popula-tion interactions. II. Destabilization processes. J. Anim. Ecol. 47:249-267.. 1979. Population biology of infectious diseases. Part II. Nature. 280:455-461.Murray, J.D. 1989. Mathematical Biology Springer-Verlag, Berlin.Myers, J.H. 1981. Interactions between western tent caterpillars and wild rose: A testof some general plant herbivore hypotheses. J. Anim. Ecol. 50:11-25.. The induced defense hypothesis: Does it apply to the poplation dynamics of insects?in Chemical Mediation of Coevolution (ed K.C. Spenser) Academic Press Inc. SanDiego. pp 345-366.^. 1988b. Can a general hypothesis explain population cycles of forest lepidoptera?Advances in Ecological Research 18:179-242.. 1990. Population cycles of Western tent caterpillars: experimental introductionsand synchrony of fluctuations. Ecology 71:986-993.Onstad, D.W. and R.I. Carruthers. 1990. Epidemiological models of insect diseases.Ann. Rev. Entomol. 35:399-419.Pielou, E.C. 1969. An Introduction to Mathematical Ecology. John Wiley & Sons, NewYork.Press, W.H., B.P. Flannery, S.A. Teukolsky and W.T. Vettering. 1988. Numerical Re-cipes in C. Cambridge Press, Cambridge. pp 387-393.Regniere, J. 1984. Vertical transmission of diseases and population dynamics of insectswith discrete generations: a model. J. Theor. Bio. 107:287-301.Rhoades, D.F. 1983. Herbivore population dynamics and plant chemistry. in VariablePlants and Herbivores In Natural and Managed Systems. (eds R.F. Denno and M.S.McClure) Academic Press, New York. pp 155-220.Smith, J.D. and R.A. Goyer. 1986. Population fluctuations and causes of mortality forthe forest tent caterpillar, Malacosoma disstria (Lepidoptera: Lasiocampidae), onthree different sites in southern Louisiana. Environ. Entomol. 15:1184-1188.Stairs, G.R. 1965. Artificial initiation of virus epizootics in forest tent caterpillars pop-ulations. Can. Ent. 97:1059-1062.Starfield, A.M. and A.L. Bleloch. 1986. Building Models for Conservation and WildlifeManagement. MacMillan Pub. Co. New York.Stenseth, N.C. 1981. On Chitty's theory for fluctuating populations: the importance ofgenetic polymorphism in the generation of regular density cycles. J. Theor. Bio.90:9-36.Thompson, W.A., I.B. Vertinsky and W.G. Wellington. 1979. The dynamics of out-breaks: further simulation experiments with the Western tent caterpillar. Res. Pop.Ecol. 20:188-200108Vezina, A. and R.M. Peterman. 1985. Tests of the role of a nuclear polyhedrosis virus inthe population dynamics of its host, douglas-fir tussock moth, Orgyia pseudotsugata(Lepidoptera: Lymantriidae). Oecologia 67:260-266.Volterra, V. 1926. Variazioni e fluttuazioni del numero d'individui in specie animali con-viventi. Mem. Acad. Lincei. 2:31-113. translated in Chapman, R.N. 1931. AnimalEcology, Appendix. pp 409-448.Wellington, W.G. 1962. Population quatlity and the maintenance of nuclear polyhedrosisbetween outbreaks of Malacosoma pluviale (Dyar). J. Insect Pathol. 4:285-305Wellington, W.G., P.J. Cameron, W.A. Thompson, I.B. Vertinsky and A.S. Landsberg.1975. A stochastic model for assessing the effects of external and internal hetero-geneity on an insect population. Res. Popul. Ecol. 17:1-28.Young, S.Y. 1990. Effect of nuclear polyhedrosis virus infection in Spodoptera ornithogallilarvae on post larval stages and dissemination by adults. J. Invert. Path. 55:69-75.109Appendix 1: ANALYSIS OF MODELSA1.1 IntroductionMany differential models and difference models can be analyzed easily by following thesame basic steps in both types of systems:1. Find the null dines and steady states of the equations.2. Linearize about each steady state and find the Jacobian and eigenvalues of theJacobian for each.3. From the eigenvalues, determine the conditions under which each steady state isst able.In this section, I analyze two of the models discussed in Chapter 2, the Lotka-Volterraequations and the Nicholson-Bailey model. All analyses are based on Edelstein-Keshet(1986).A1.2 Lotka-VolterraThe Lotka-Volterra model (equations 3.6) can be analyzed using the steps mentionedabove.1. Find and graph the null dines and steady states of the equations:• Null dines: These are found by setting the left-hand side of each of the equa-tions to zero. So equation 3.6a becomes:which implies:Equation 3.6b gives:which implies0 rN — bPNN= 0 or P r lb.0 cPN — dP(A1.1)(A1.2)(A1.3)P= 0 or N dlc^(A1.4)• Steady states: These are found wherever a null dine from each of equationsA1.2 and A1.4 cross (but not where two null dines from the same equationcross). For the Lotka-Volterra equations, there are two such equilibria:trivial:^(N,P) = (0,0)non-trivial: (N ,P)^(d1c,r lb)2. Find the Jacobian and eigenvalues for each of the steady states:• at (0,0): Jacobian:^rr^0 1(A1.5)[0 —d •The eigenvalues are: r and —d. Since one eigenvalue is greater than zero andone is less than zero, (0, 0) corresponds to a saddle point.• at (d1c,r1b): Jacobian:[ 0^—bd/c1cr lb^0 (A1.6)[ _I'110Here the eigenvalues are: +i\/rd. Notice that the real part of the eigenvaluesis 0, so the equilibrium is neutrally stable with oscillations about it.A1.3 Nicholson-BaileyNext, the Nicholson-Bailey model (equations 3.11) for insect-parasitiod systems will beanalyzed following the steps above.1. Null Clines and Steady StatesThe null dines of difference equations can be found, by definition, when the pop-ulation is not changing or Nt+i Nt . Thus, solving for the isoclines in equation3.11a leads to the conditions:N = 0 or P ln rawhile equation 3.11b gives the isocline:Pc(1 — e — aPt)Thus, there are two steady states:(N, P) (0,0)and^—D\^rinr^lnr \\Iv^)^ac(r — 1)^a ).(A1.7)(A1.8)(A1.9)(A1.10)A1.9, is the trivial equilibrium while equation A1.10 is them.e— a15^—aN e — a lc(1 — e — a 1-5 ) cal Tre — a15[When evaluated at the trivial equilibrium (A1.9), matrix A1.11 becomes:[Or 01^(A1.12)Clearly the eigenvalues of this matrix are r and 0. In difference equations, a steadystate is stable if all the eigenvalues of the Jacobioan evaluated at that steady stateare between -1 and 1. Thus, in this case, the trivial equilibrium is stable if r <1. This means that the population will die out if r < 1. The second case ismore complicated. When the Jacobian is evaluated at the non-trivial equilibrium(A1.10), matrix A1.11 becomes:r In r c(r —1)In r •(r-1)N(A1.13)The first, equationnon-trivial eqilibriu2. The JacobianThe general matrix is( A 1 .11 )111So, letln ri3 = 1 + ^ (A1.14)(r ^1)and,r In r7 =^^(r — 1)^• (A1.15)Now the stability condition given by Edelstein-Keshet (1986) for difference equa-tions, 2 < 1 + -y < 1 /31, can be checked. It is easy to show (Edelstein-Keshet, p82,1986) that the first half, 2 < 1 + -y, is true only if r < 1 while the second part,1+7 < 101, holds only if r > 1. Therefore, the non-trivial equilibrium is never stable(and if the trivial equilibrium is also unstable, the population will show increasing,unstable, oscillations).112Appendix 2: ANALYSIS OF THE MODELA2.1 IntroductionIn this appendix, I provide the details for the analysis of the new model (equations 5.1).In the first part, I derive an approximate reproductive rate for the virus. In the secondpart, since the techniques for analyzing systems of equations are already mentionedin Appendix 1, the equations are simplified and are put into a form for which thesetechniques can be used, and the stability of the system determined. The third sectiondescribes some of the techniques that I used to attempt to numerically determine thestability of the system and the possible existence of cycles.A2.2 Virus reproductive rateThe first thing that can be calculated is the reproductive rate for the virus. Aftersubstituting equation 5.1c into equation 5.1d, the basic equation for the virus becomes:Vy ,t+1 = Cr y Vy ,t^(rt^wt )Pt ( 0-3 Sy,t^(7e Ey ,t)^(A2.1)where, as before,pt = 1 — (1 +  Vt^(5.3)Ow t Hy , tEquation A2.1 is non-linear, due to the form of P t . This makes any analysis difficult.Therefore, to simplify the analysis, pt can be linearized about V = 0 to get:Vt Pt tot Hy , tSubstituting A2.2 into A2.1,1Vt+1^7v V, ,t^ Wt)(C r S y,t^aeEy ,t)Wt H y,tand rearranging the terms gives:Vy,t+1^RtVy,teR =^(ytq Wt)(6rsSy,t UeEy,t) where,WtHy,t(A2.2)(A2.3)(A2.4)(A2.5)Hy,t = Sy ,t^Ey ,t^,t^ (A2.6)Since Pt was linearized about V = 0, A2.4 is valid only when the amount of virus issmall.In theory, it would be possible to find other expressions for the reproductive rateof the virus that would be accurate at different amounts of virus. This would be doneby linearlizing p about some other steady state (where none of the populations arechanging). In practice, this is not possible analytically without making other majorsimplifications to the model.113A2.3 Equilibrium AnalysisTo determine analytically the behaviour of the system near equilibrium the model mustfirst be simplified. Since it was the term pt , that was creating the difficulties in analysis,I simplified the model by assuming that p remained constant. The analysis can thenbe easily performed on the adult stage of the model, by combining all tents. Therefore,merging equations 5.8-5.10, the total numbers of organisms are:Sy+i,o =2(f8Sy ,s+ (1 — i)feEy,6)^(A2.7a)dy r—Kj_Gy , 6^(A2.7b)2Ty+1 =2 Y'6 + Ey ,6(A2.7c)and, as before, d is given bydy^1 (5.7)a + TyIt follows from the assumptions in the model that the number of adults in year y isdependent on the number of young starting the same year, and, due to the simplification,that the dependence is linear. Thus,Sy,6^XSy,0^ (A2.8a)Ey,6 = ZSy,0 (A2.8b)where x and z are independent of the state variables. (The exact formulas for x and zare not important for the stability calculations but are shown in equations A2.19 andA2.22 and discussed further at that place.) Substituting equations A2.8 into equationsA2.7 and dropping the t subscript since t = 0 on both sides of the equation, gives:dySy+1 = —2 (Lx + (1 — n)fe z)Sy^(A2.9a)/y+1 = —dy KfezSy (A2.9b)2dyTy+1 = —2 (x z)Sy .^ (A2.9c)Using the standard techniques demonstrated in Appendix 1.3 (step 1), the equilibriumvalues for this system (equations A2.9) can be found at:(S, I, T) = (0, 0, 0)^ (A2.10)and at,where,S = ^— 2)2(x z)/ = Kfe za(0 — 2)2(x + z)alkT2 a(A2.11)4+1,0 =114f8 x + (1 — ti)fe z.^ (A2.12)The term represents the amount of reproduction of the organisms.The second equilibrium, given by equations A2.11, does not exist if density-depend-ence was not present. To see this, the density-dependence term dy in equations A2.9would be replaced with the constant 1. If the equilibrium of A2.9 was calculated asbefore, the only possible steady state would be the trivial one given in equation A2.10.The Jacobian for the system (equations A2.9) iscrcPJ2(ce-I-T)atcf, z02(a+T) 2aKf2z 3 (A2.13)2(cr+T)cx(x+z)2(a-FT) 2a(x+ z)2(a+T) 2(a-FT) 2^_Substituting the trivial equilibrium (equation A2.10) into matrix A2.13, and calculatingthe eigenvalues of the resulting matrix, it can be seen that the eigenvalues are 0, 0, andIk/2. Similarily, the eigenvalues of the Jacobian evaluated at the non-trivial equilibrium(A2.12) are 0, 0 and 2/0. In Appendix 1.3, it was noted that if Ai is an eigenvalue, thecondition for stability of an equilibrium is that 'Ail < 1 for all i. Clearly, 0 < 1, so thestability of each eigenvalue depends on the value of 0. There are two possibilities:< 2 A2.10 stable (A2.14)> 2 A2.11 stable (A2.15)Cycles are not possible in either case. As 7/, approaches 2 from above, the non-trivialequilibrium point appraoches the trivial one, until, when = 2, the non-trivial equi-librium equals the trivial one. For cycles to be possible, either both equilibria mustbe saddle-nodes, in which case a saddle node connector cycle may be present, or atleast two of the eigenvalues of a steady state must be 'Ail = 1, in which case a Hopfbifurcation leading to limit cycles may be present (Guckenheimer and Holmes, 1983).The virus V can also be included in the analysis. Following the same simplificationprocess that was done above to change equations A2.8 into A2.10, the amount of virusfrom year to year isV +1 = a6Vy coSY ZSy (A2.16)where w is some combination of parameters only, and is independent of all the variables.(As with x and z, the exact form of w is not important to the stability calculations.)When the virus (in the form of equation A2.16) is included in the analysis, the onlychange in the results is that the resulting Jacobian has another eigenvalue that equals. Since this is always less than one, the stability conditions given in equations A2.14and A2.15 do not change.The parameter p was assumed to be constant for the purposes of doing this analysis.However, the value of p affects the results. To see what happens as p is varied, the forms115for x and z must be calculated. They are found by solving the difference equations forthe within-year dynamics.Syt+i = Q3 (1 — 13)S y^ (5.1a)becomesSy , t = (u3 (1 — p)) t Sy , o . (A2.17)Equation 5.1a could not be solved explicitly if p were dependent on any of the othermodel variables. Since there were six time-steps for this part of the model, let t = 6.Also, the survival from the pupal stage must be included. So, equation A2.17 becomesSy,6 = (us (1 — p)) 6 up Sy ,0^ (A2.18)This is now in the form of equation A2.9a so clearlyx = (o-3 (1 — p)) 6 gyp .^ (A2.19)To calculate z, the same procedure is followed, using equation A2.17, and rememberingthat no organisms are born exposed (Ey , 0 0). Equation 5.1b can be rearranged togiveEy ,t+i = p(1 — q)usSy,t (1 — Pg)creEy ,t •This then becomesEy ,t^Cris (1 — py 11 \QS (1 — 1)) — (1 — pq)u e 13(1 —^S y(A2.20)(A2.21)After allowing for pupal mortality, and noticing that equation A2.21 is in the same formas equation A2.9b, it is clear thatz =^(1 — p) 6 —(1 — pq) 6 0 6'78( 1 — — (1 — pq ) cr e) P( 1 0 0-sup.(A2.22)Upon examination of x and z given in equations A2.19 and A2.22 and shown in Figure18, it can be seen that x > z for small p (since all parameters involved in equationsA2.19 and A2.22 are < 1). As p approaches one (its upper bound), x goes to 0 and zbecomes small. Since 7b depends on x and z, it will also be small, less than 2 (unlessfe z the birth-rate of exposed mothers is very high). As p becomes small, lk will grow.In general, there is a critical value, pc which corresponds to the case 7b = 2. If p > pcthen the population will die out while if p < pc the non-trivial equilibrium is stable (seeFigure 17). There is no explicit formula for pc . Thus, the value of p that is chosen affectsthe level at which the population stabilizes, and even whether or not the population willsurvive. Obviously, the other parameters involved in will also help determine whatvalue has. Similar analysis could be done for the other parameter values, but sincethe results from these other analysis would also be dependent on p having a constantvalue, and would have less meaning in the full model, the analysis will not be presented.1160.0^ 0.2^0.4^0.6^0.8^1.0Infection probability (p)Figure 18 Parameters x and z versus the infection probability, p. Parameters are:a, = .93, a, .87 and q = .5 (the base values given in Table 5).117A2.4 Stability: numerical analysisSince the general analysis performed in section A2.3 is necessarily inaccurate due tothe simplification that was made, I also attempted some numerical analysis of the fullmodel. As with the previous analysis, if a steady state can be found, the eigenvaluesof the Jacobian of the matrix will describe the behaviour of the populations near thesteady state. One way for cycles to occur is to have a Hopf bifurcation. The firstcriterion for finding such a bifurcation is that for some set of parameters there mustbe at least one pair of complex conjugate eigenvalues with lA i I = 1 (Guckenheimer andHolmes, 1983). The bifurcation will arise at this point.In order to find the eigenvalues, it was first necessary to find a steady state and tocalculate the Jacobian at this point. Initially, I found a steady state by simulating themodel until the population stabilized (the population size no longer varied). I then tookthe partial derivatives of the model at this steady state and created a Jacobian matrix.Next, I used an eigenvalue finding program (Press et. al., 1988, pp 387-393) to find theeigenvalues of the Jacobian.In practice, there was a major problem with computer generated error. The methodthat I used to find the eigenvalues stated that the error would be on the order of theeuclidean norm of the Jacobian. This was quite large (sometimes up to 80%). In orderto find a Hopf bifurcation, it is important to know when the eigenvalues change fromless than one to greater than one. Having large error makes it impossible to find thispoint.The was evident from the simulations that I did. For example, if the steady stateis stable, the magnitudes of the eigenvalues should all be less than one. The method Iinitially used to find the steady state (by running the model on a computer until thepopulation size no longer changed) guaranteed that the steady state was a stable one.Simulations perturbing the population away from the steady state confirmed this sincethe population always returned within three generations. However, the eigenvalues thatwere found were not all less than one. If the true value of an eigenvalue was near one,even a small amount of error could be enough to make it larger than one, or to createa complex value when the true value was real. This error made this technique uselessand no further numerical analysis was performed using this or any other method.118


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