Patterns in a Modified Leslie–Gower Model with Beddington–DeAngelis Functional Response and Nonlocal Prey Competition

Abstract

In this paper, we present the theoretical results on the pattern formation of a modified Leslie–Gower diffusive predator–prey system with Beddington–DeAngelis functional response and nonlocal prey competition under Neumann boundary conditions. First, we investigate the local stability of homogeneous steady-state solutions and describe the effect of the nonlocal term on the stability of the positive homogeneous steady-state solution. Lyapunov–Schmidt method is applied to the study of steady-state bifurcation and Hopf bifurcation at the interior of constant steady state. In particular, we investigate the existence, stability and multiplicity of spatially nonhomogeneous steady-state solutions and spatially nonhomogeneous periodic solutions. Furthermore, we present a simple description of the dynamical behaviors of the system around the interaction of steady-state bifurcation curve and Hopf bifurcation curve. Finally, a numerical simulation is provided to show that the nonlocal competition term can destabilize the constant positive steady-state solution and lead to the occurrence of spatially nonhomogeneous steady-state solutions and spatially nonhomogeneous time-periodic solutions.

Global dynamics and spatio-temporal patterns in a two-species chemotaxis system with two chemicals

“…This paper takes a step forward to present the nonconstant steady states for (1.1), and our numerical simulations show that (1.1) can admit complex patterns and rich dynamics in some special signal production mechanisms. However, a big challenging question for (1.1) is how to justify the existence of nonconstant steady states (see [7,19,24,31,40,52]) even for the following system, which can be regarded as a special case of (1.…”

Stability analysis of a stage‐structure model with spatial heterogeneity

“…Moreover, the spatially non-homogeneity of the parameters in system (2) makes it impossible to discuss (2) as in the previous literature. [22][23][24][25][26][27][28][29] In Lou 30 and Cantrell and Cosner, 31 the existence of a positive steady-state solution, as well as the effects of dispersal and spatial heterogeneity, is obtained for a logistic population model with a function m(x) representing the intrinsic growth rate of the species. The main reason for the introduction of the spatial variation in m(x) is to investigate how the favorable (i.e., m(x) > 0) and unfavorable (i.e., m(x) < 0) habitats affect the dynamics of population.…”

Persistence and extinction of a stochastic SIS epidemic model with regime switching and Lévy jumps

“…Throughout this paper, we always assume that lim S→0 g(S, I)/S exists for all I ≥ 0, g (S, 0) lim I→0 g(S, I)/I > 0 for all S > 0, and g(S, I)/I is monotone nonincreasing with respect to I ∈ (0, ∞) for S ∈ (0, ∞). Obviously, the function g includes some special incidence rates [10,11,15,17,25,26,40,41], such as bilinear incidence rate g(S, I) = SI, saturation incidence rate g(S, I) = SI/(1 + mI) with a positive constant m denoting the half-saturation constant, Holling type II incidence rate g(S, I) = SI/(1 + mS), Beddington-DeAngelis incidence rate g(S, I) = SI/(aS + bI + c) with positive constants a, b, and c, and some other kinds of incidence rates like g(S, I) = e −mI SI and g(S, I) = SI/(1 + mI θ ) with positive constants m and θ (see, for example [8,30,36]). In [8], model (1) with bilinear incidence rate always has a globally asymptotically stable disease-free equilibrium E 0 and so the disease disappears if the basic reproduction number R 0 = Λβ µ(µ+α+λ) ≤ 1.…”