Oscillations and Pattern Formation in a Slow-Fast Prey-Predator System

Evolution of Cooperation in Spatio-Temporal Evolutionary Games with Public Goods Feedback

In biology, evolutionary game-theoretical models often arise in which players' strategies impact the state of the environment, driving feedback between strategy and the surroundings. In this case, cooperative interactions can be applied to studying ecological systems, animal or microorganism populations, and cells producing or actively extracting a growth resource from their environment. We consider the framework of eco-evolutionary game theory with replicator dynamics and growth-limiting public goods extracted by population members from some external source. It is known that the two sub-populations of cooperators and defectors can develop spatio-temporal patterns that enable long-term coexistence in the shared environment. To investigate this phenomenon and unveil the mechanisms that sustain cooperation, we analyze two eco-evolutionary models: a well-mixed environment and a heterogeneous model with spatial diffusion. In the latter, we integrate spatial diffusion into replicator dynamics. Our findings reveal rich strategy dynamics, including bistability and bifurcations, in the temporal system and spatial stability, as well as Turing instability, Turing-Hopf bifurcations, and chaos in the diffusion system. The results indicate that effective mechanisms to promote cooperation include increasing the player density, decreasing the relative timescale, controlling the density of initial cooperators, improving the diffusion rate of the public goods, lowering the diffusion rate of the cooperators, and enhancing the payoffs to the cooperators. We provide the conditions for the existence, stability, and occurrence of bifurcations in both systems. Our analysis can be applied to dynamic phenomena in fields as diverse as human decision-making, microorganism growth factors secretion, and group hunting.

Allee effect-driven complexity in a spatiotemporal predator-prey system with fear factor

In this paper, we propose a spatiotemporal prey-predator model with fear and Allee effects. We first establish the global existence of solution in time and provide some sufficient conditions for the existence of non-negative spatially homogeneous equilibria. Then, we study the stability and bifurcation for the non-negative equilibria and explore the bifurcation diagram, which revealed that the Allee effect and fear factor can induce complex bifurcation scenario. We discuss that large Allee effect-driven Turing instability and pattern transition for the considered system with the Holling-Ⅰ type functional response, and how small Allee effect stabilizes the system in nature. Finally, numerical simulations illustrate the effectiveness of theoretical results. The main contribution of this work is to discover that the Allee effect can induce both codimension-one bifurcations (transcritical, saddle-node, Hopf, Turing) and codimension-two bifurcations (cusp, Bogdanov-Takens and Turing-Hopf) in a spatiotemporal predator-prey model with a fear factor. In addition, we observe that the circular rings pattern loses its stability, and transitions to the coldspot and stripe pattern in Hopf region or the Turing-Hopf region for a special choice of initial condition.

Analytical detection of stationary and dynamic patterns in a prey-predator model with reproductive Allee effect in prey growth

Allee effect in population dynamics has a major impact in suppressing the paradox of enrichment through global bifurcation, and it can generate highly complex dynamics. The influence of the reproductive Allee effect, incorporated in the prey's growth rate of a prey-predator model with Beddington-DeAngelis functional response, is investigated here. Preliminary local and global bifurcations are identified of the temporal model. Existence and non-existence of heterogeneous steady-state solutions of the spatio-temporal system are established for suitable ranges of parameter values. The spatio-temporal model satisfies Turing instability conditions, but numerical investigation reveals that the heterogeneous patterns corresponding to unstable Turing eigenmodes act as a transitory pattern. Inclusion of the reproductive Allee effect in the prey population has a destabilising effect on the coexistence equilibrium. For a range of parameter values, various branches of stationary solutions including mode-dependent Turing solutions and localized pattern solutions are identified using numerical bifurcation technique. The model is also capable to produce some complex dynamic patterns such as travelling wave, moving pulse solution, and spatio-temporal chaos for certain range of parameters and diffusivity along with appropriate choice of initial conditions. Judicious choices of parametrization for the Beddington-DeAngelis functional response help us to infer about the resulting patterns for similar prey-predator models with Holling type-II functional response and ratio-dependent functional response.

Effect of Slow–Fast Time Scale on Transient Dynamics in a Realistic Prey-Predator System

Systems with multiple time scales, often referred to as `slow–fast systems’, have been a focus of research for about three decades. Such systems show a variety of interesting, sometimes counter-intuitive dynamical behaviors and are believed to, in many cases, provide a more realistic description of ecological dynamics. In particular, the presence of slow–fast time scales is known to be one of the main mechanisms resulting in long transients—dynamical behavior that mimics a system’s asymptotic regime but only lasts for a finite (albeit very long) time. A prey–predator system where the prey growth rate is much larger than that of the predator is a paradigmatic example of slow–fast systems. In this paper, we provide detailed investigation of a more advanced variant of prey–predator system that has been overlooked in previous studies, that is, where the predator response is ratio-dependent and the predator mortality is nonlinear. We perform a comprehensive analytical study of this system to reveal a sequence of bifurcations that are responsible for the change in the system dynamics from a simple steady state and/or a limit cycle to canards and relaxation oscillations. We then consider how those changes in the system dynamics affect the properties of long transient dynamics. We conclude with a discussion of the ecological implications of our findings, in particular to argue that the changes in the system dynamics in response to an increase of the time scale ratio are counter-intuitive or even paradoxical.