Stationary patterns induced by self- and cross-diffusion in a Beddington–DeAngelis predator–prey model

Global dynamics of a diffusive SIR epidemic model with saturated incidence rate and discontinuous treatments

In this paper, we study a diffusive SIR epidemic model with saturated incidence rate and discontinuous treatments under Neumann boundary conditions. Firstly, the existence and boundedness of the solution of the system are addressed. Then, on the basis of the differential inclusions theory, we analysis the existence of endemic equilibrium. Furthermore, by constructing different appropriate Lyapunov functions, we investigate the global asymptotic stability of the disease free equilibrium(DFE) and the endemic equilibrium(EE), respectively. Additionally, numerical simulations are given to confirm the correctness of theorem. Finally, we give a brief conclusion and discussion in the end of the paper.

Appearance of Temporal and Spatial Chaos in an Ecological System: A Mathematical Modeling Study

The ecological theory of species interactions rests largely on the competition, interference, and predator-prey models. In this paper, we propose and investigate a three-species predator-prey model to inspect the mutual interference between predators. We analyze boundedness and Kolmogorov conditions for the non-spatial model. The dynamical behavior of the system is analyzed by stability and Hopf bifurcation analysis. The Turing instability criteria for the Spatio-temporal system is estimated. In the numerical simulation, phase portrait with time evolution diagrams shows periodic and chaotic oscillations. Bifurcation diagrams show the very rich and complex dynamical behavior of the non-spatial model. We calculate the Lyapunov exponent to justify the dynamics of the non-spatial model. A variety of patterns like interference, spot, and stripe are observed with special emphasis on Beddington-DeAngelis function response. These complex patterns explore the beauty of the spatio-temporal model and it can be easily related to real-world biological systems.