From local interactions to population dynamics in site-based models of ecology

Abstract

A central problem in ecology is relating the interactions of individuals-described in terms of competition, predation, interference, etc.-to the dynamics of the populations of these individuals-in terms of change in numbers of individuals over time. Here, we address this problem for a class of site-based ecological models, where local interactions between individuals take place at a finite number of discrete resource sites over non-overlapping generations and, between generations, individuals move randomly between sites over the entire system. Such site-based models have previously been applied to a wide range of ecological systems: from those involving contest or scramble competition for resources to host-parasite interactions and meta-populations. We show how the population dynamics of site-based models can be accurately approximated by and understood through deterministic and stochastic difference equations. Conversely, we use the inverse of this approximation to show what implicit assumptions are made about individual interactions by modelling of population dynamics in terms of difference equations. To this end, we prove a useful and general theorem: that any model in our class of site-based models has a corresponding stochastic difference equation population model, by which it can be approximated. This theorem allows us to calculate long-term population dynamics, evolutionary stable strategies and, by extending our theory to account for large deviations, extinction probabilities for a wide range of site-based systems. Our methodology is then illustrated to various examples of between species competition, predator-prey interactions and co-operation. r

Stochastic analogues of deterministic single-species population models

“…In the case where there is no environmental noise Johansson & Sumpter (2003) have shown that the population dynamics of the model are well-approximated by the stochastic dynamical system…”

Costly dispersal can destabilize the homogeneous equilibrium of a metapopulation

“…We know relatively little about the underlying mechanisms that create the discrete-time maps (see e.g. Gyllenberg et al 1997;Gamarra and Sole 2002;Johansson and Sumpter 2003;Thieme 2003;Geritz and Kisdi 2004;Eskola and Geritz 2007;Eskola and Parvinen 2007 for mechanistic underpinnings of various discrete-time models), and therefore we don't have a priori constraints on the convexity of f. Empirical data are usually too noisy to ascertain the precise shape of f. There is thus no ground for narrowing research to the few famous discrete-time population models. As the present study also underlines, it is dangerous to overuse just a few models, because results based on them may not carry over to other models.…”

Kappa Function as a Unifying Framework for Discrete Population Modeling

“…If P is the size of the parasite, then the probability of the host escaping parasitism is given by (2) with x = γ · P , where γ is the attack rate of the parasitoid, and κ is the aggregate parameter of the parasitoid. In addition to parasitism, and individual fitness, the framework of kappa function may include the Allee effect phenomenon as was pointed out by Johansson and Sumpter [2003]. This may be done by requiring, for example, that two or more individuals are needed per resource site for offspring to be produced.…”