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Ecosystem dynamics can be studied using model populations. Perturbation experiments have often been applied to simulated populations in order to study the uncertainty in ecosystem dynamics. Most of these studies have predicted a stationary state. We report on the uncertainty in the dynamics near extinction. In particular, we explore fluctuation enhancements, i.e., enhanced variability in dynamics of paths to extinction. We examine two dynamic models on a two-dimensional lattice of finite size: (1) the contact process (CP) in which interactions are restricted to occur between adjacent lattice points, and (2) mean-field simulation (MFS), where interactions occur globally, between any pair of lattice points. Computer simulation reveals that, for both CP and MFS, the random drift of density about a stationary state increases with the decrease of steady-state density. Drift is much more pronounced in the vicinity of the critical mortality rate, at the transition to extinction. Simulation demonstrates that MFS shows greater variability when relative mortality rate is low whereas CP shows much more pronounced variability when mortality is near the critical threshold. The CP process shows wider fluctuations while the MFS process shows minimal increases following perturbations that lead to extinction. Because interactions are local for CP, there are a variety of different paths to extinction.

There has been much literature on ecological model of Prisoner's Dilemma (PD) game. This game illustrates that cooperation can evolve in situations where individuals tend to look after themselves. In order to explain some behaviors of altruism in animal societies, the strategy All Cooperate (AC), often called the Golden Rule, is more appropriate than other strategies. However, very little is known about the superiority of AC. In the present article, we study patch dynamics based on non-iterated PD game, applying two different methods: island and lattice models. Each patch is assumed to be either vacant or composed of a population of AC or All Defect (AD), where AD means a selfish strategy. Both models exhibit a phase transition between a phase where both AC and AD survive, and a phase where AD is extinct. The latter phase means that AC beats AD completely. In the case of lattice model, the extinction of AD easily occurs and the abundance of AC takes a larger value, compared with the island model. Our models can be also extended to general iterated PD game; we describe the reason why AC can outperform any other strategy.