Approaches to Modelling a Predator-Prey System in 2 D Space

We compare two approaches to simulating predator-prey dynamics with spatial e ects: as an agentbased system, and as a variant of reaction-di usion. As a system of agents, we observe that rare predator success and slow predator respone to an increase in prey numbers both reduce the magnitude of oscillations, reducing the chances of an extinction. Apart from the consequences of discretization error (such as extinctions) and of a stochastic component in predator and prey growth rates, the agent-based model agrees with the Lotka-Volterra model of population dynamics. In our PDE-based reaction-di usion model, oscillations initially occur over space instead of time. These spatial oscillations disappear once the space is saturated; the subsequent temporal oscillations are dampened over time, especially by xed boundary conditions and/or self-limitation of prey, and eventually disappear. 1 Background We will investigate predator-prey relationships in terms of population dynamics, additionally introducing spatial e ects. In so doing, we seek to better understand the trophic web (generalizing the notion of 'food chain'), and thereby its parent ecology. Moreover, we will explore two distinct approaches to simulating group dynamics by treating species in terms of individual agents, or as quantities in ux. We begin with the ODE (ordinary di erential equation) model of population dynamics alone. We let the growth rate of the prey increase with the prey population (initially, at least) and decrease with the # of prey-predator interactions. We let the growth rate of the predator population increase with said # of interactions and decrease with the predator population (i.e. competition leading to self-limitation). Algorithm 1 Lotka-Volterra predator-prey equations, with self-limitation of prey. du dt = αu (1− u)− βuv dv dt = βuv − γv u, v are functions of time, t. α, β, γ are positive constants. We require that u(t), v(t) ∈ [0, 1]∀t. Observe that u (1− u) is positive if and only if u < 1, and so the carrying capacity of species u is 1. When the initial population of both species is small, prey numbers tend to rise, followed by predator numbers leading to a crash in prey numbers, followed by a crash in predator numbers, and then the cycle repeats. In the absence of di usion, this leads to oscillatory behavior in time. Observe also that whilst the prey can persist without the predator, the predator cannot persist without the prey. We will discuss both some results from and the technical tradeo s of an agent-based and PDE-based approach to implementing spatial e ects. PDEs (partial di erential equations) generalize ODEs to multiple dimensions (e.g. 2d space in addition to time), whereas agents are de ned in terms of individuals' behavior rather than the more abstract notion of a population's behavior. In an agent-based system implemented in Repast (an agent-based modelling framework), with discrete population and stochastic predator-prey interactions, the model is prone to extinction events when 'downswings' are severe (i.e. the amplitude of the oscillations is large).