UBC Theses and Dissertations
Scaling, survival and extinction in many-species systems Ifti, Margarita
In this thesis we study the behaviour of three- and many-species systems, when the stochastic nature of the interactions within them is taken into consideration. We pay special attention to the appearance of cooperative states in them. We also study a model of development of cooperation, called Continuous Prisoner's Dilemma. We start with the cyclic ABC model (A + B → 2B, B + C → 2C, C + A → 2A), and its counterpart: the three-component neutral drift model (A + B → 2B or 2A, B + C → 2C or 2B, C + A → 2A or 2C). In the former case, the mean-field approximation exhibits cyclic behaviour with an amplitude determined by the initial condition. When stochastic phenomena are taken into account, the amplitude of the oscillations drifts and eventually one and then two of the species go extinct. The second model remains stationary for all initial conditions in the mean-field approximation, and drifts when stochastic phenomena are considered. We analyzed the distribution of first extinction times of both models by simulations of the master equation, and from the point of view of the Fokker-Planck equation. Survival probability vs. time plots suggest an exponential decay. For the neutral model the extinction rate is inversely proportional to the system size, while the cyclic system exhibits anomalous behaviour for small system sizes. In the large system size limit the extinction times for both models will be the same. This result is compatible with the smallest eigenvalue obtained from the numerical solution of the Fokker-Planck equation. We also studied the behaviour of the probability distribution. For evolutionary times, it becomes uniform, which is in agreement with the exponential decay. The exponential behaviour is found to be robust against certain changes, such as the three reactions having different rates. Finally, we study the system with an intermediate selection coefficient, and show that for evolutionary times it behaves like the cyclic (deterministic) and neutral drift models. When mutation or migrations are taken into consideration, there are three distinct regimes in the model, (i) In the "fixation" regime, the first extinction time scales with system size N and has exponential distribution with an exponent that depends on the mutation/migration probability µ. (ii) In the "diversity" regime, the system accepts the linear noise approximation, and exhibits Ornstein-Uhlenbeck process behaviour. All three species are present in the system at almost all times.' The analytical results are checked against computer simulations of the master equation, (iii) In the critical regime, the first "extinction time has a power-law distribution with exponent -1. The transition corresponds to a crossover from diffusive behaviour to Gaussian fluctuations around a stable solution. The critical µ is written as µ₀/N and µ₀ is determined from the simulations, as well as numerical solution of a nonlinear- Fokker-Planck equation to be about 0.33. The above-described behaviour is observed for systems with four- or more species, and a possible mechanism for the appearance of symbiotic pairs is discussed. The model is used for emulating a computer network with e-mail viruses. It is observed that clustering is necessary for pandemics to happen. We discuss ' differences between computer users, e-mail, and social network topologies and their role in determining the nature of epidemics. The evolution of cooperation is studied, using the Continuous Prisoner's Dilemma. This is a Prisoner's Dilemma in which the cooperation is not all or nothing, but rather goes through baby steps, from small to bigger degrees of assistance. It has been shown that in the presence of spatial structure the individual's investment reaches substantial levels, and fluctuates around those. We examine the effects of increasing the neighbourhood size, and study the limits at which the mean-field behaviour of pure defection becomes the prevailing one. It is observed that when the neighbourhood sizes for "playing against" and "comparing with" differ by more than 0.5, the cooperative behaviour is unsustainable. Further, neighbourhood size mutations are introduced, and the parameters are treated as phenotypes. The final state is a cooperative one, and the neighbourhood and dispersal parameters values are observed to converge toward one-another. When placed in a network, it is observed that the average degree of the network for which cooperation is still sustained practically does not depend on network size, and is much larger in the clustered social networks than in the distributed random networks. This further strengthens the argument that clustered spatial structures help the development and maintenance of cooperation.
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