Exploring Spatiotemporal Complexity of a Predator‐Prey System with Migration and Diffusion by a Three‐Chain Coupled Map Lattice

Abstract

The topic of utilizing coupled map lattice to investigate complex spatiotemporal dynamics has attracted a lot of interest. For exploring the spatiotemporal complexity of a predator-prey system with migration and diffusion, a new three-chain coupled map lattice model is developed in this research. Based on Turing instability analysis, pattern formation conditions for the predator-prey system are derived. Via numerical simulation, rich Turing patterns are found with subtle self-organized structures under diffusion-driven and migration-driven mechanisms. With the variation of migration rates, the predator-prey system exhibits a gradual dynamical transition from diffusion-driven patterns to migration-driven patterns. Moreover, new results, the self-organization of non-Turing patterns, are also revealed. We find that even in the cases where the nonspatial predator-prey system reaches collapse, the migration can still drive pattern self-organization. These non-Turing patterns suggest many new possible ways for the coexistence of predator and prey in space, under the effects of migration and diffusion.

Geometric Numerical Integration in Ecological Modelling

“…When space is included in the models, the formation of spatial patterns constitutes a very active research area in ecosystem modelling, based on the pioneering work of Turing [27]. There is a huge literature on the subject; however, to show the benefits of using Poisson integrators for predicting the onset of Turing patterns here we will focus on the specific model described in [28]. It is a three-chain coupled map lattice model built for exploring the spatiotemporal complexity of a predator-prey system with migration and diffusion.…”

“…In [28] the discrete reaction semiflow advances in time with explicit Euler. Here, we use Poisson Euler scheme to investigate the performance of a geometric numerical integrator.…”

“…We use the forward-backward sweep approach described in [40] joint with an exponential Lawson symplectic scheme [39]. This choice is motivated by the fact that, for all x ∈ Ω h the ODE system (28) and (29) is what is referred in literature as a nearly Hamiltonian system :…”

“…In more details, given a Runge-Kutta scheme defined by coefficients a i,j , b i , with b i = 0 and its symplectic counterpart defined by coefficientsâ i,j , b i , with b k b j = b kâkj + b j a jk for each k, j and c k =ĉ k = ∑ s j=1âk,j with c s = 1, the scheme applied to (28) and (29) proceeds according to the following steps:…”

Dynamic Analysis of a Heterogeneous Diffusive Prey-Predator System in Time-Periodic Environment

“…Especially, many biologists argued that the ratio-dependent theory is more suitable for describing prey-predator systems in many situations [13,[30][31][32]. Since Ardini and Ginzburg proposed the ratio-dependent prey-predator system, the prey-predator systems with ratio-dependent functional response are widely studied [13,[33][34][35][36], and many interesting results are obtained.…”