Stability and bifurcation on predator-prey systems with nonlocal prey competition

Abstract

In this paper, we investigate diffusive predator-prey systems with nonlocal intraspecific competition of prey for resources. We prove the existence and uniqueness of positive steady states when the conversion rate is large. To show the existence of complex spatiotemporal patterns, we consider the Hopf bifurcation for a spatially homogeneous kernel function, by using the conversion rate as the bifurcation parameter. Our results suggest that Hopf bifurcation is more likely to occur with nonlocal competition of prey. Moreover, we find that the steady state can lose the stability when conversion rate passes through some Hopf bifurcation value, and the bifurcating periodic solutions near such bifurcation value can be spatially nonhomogeneous. This phenomenon is different from that for the model without nonlocal competition of prey, where the bifurcating periodic solutions are spatially homogeneous near such bifurcation value.

Almost periodicity analysis for a delayed Nicholson's blowflies model with nonlinear density-dependent mortality term

“…Introduction. Many problems in the fields of physics [1,2], mathematical biology [3,4,5,6,7,8,9] and control theory can be attributed to study of the nonlinear differential equations, especially it is almost periodicity because there is almost no phenomenon that is purely periodic [10,11,12]. Consequently, the qualitative theory of differential equations involving almost periodicity has been the new world-wide focus.…”

Pattern formation of a diffusive predator‐prey model with herd behavior and nonlocal prey competition

“…l (see, for instance, previous works 28,32,36 ) the system (2) becomes in the presence of the nonlocal term…”

“…In Chen and Shi, it has been revealed that the most straightforward way for including nonlocal impact is to replace in the system by , where ϕ ( x , y ) is reasonable kernel. Choosing (see, for instance, previous works) the system becomes in the presence of the nonlocal term …”

“…It is well known that the competition of the prey at a locative location can not only depend on the locative prey density but also depend on the average of the thereabout prey. The effect of the nonlocal competition has been studied widely in different cases, as the single species, 26 predator-prey interaction, 14,[27][28][29][30][31][32] competing populations, 33 and so on. For instance, the papers 14,32,34 and many other papers deal with the presence of the nonlocal competition in a prey population.…”

Spatial Nonhomogeneous Periodic Solutions Induced by Nonlocal Prey Competition in a Diffusive Predator–Prey Model

“…Recently, considering nonlocal competition of prey, Merchant and Nagata [20] proposed the following nonlocal When Ω = (−∞, ∞), they showed that the nonlocal competition can induce complex spatiotemporal patterns. For one-dimensional bounded domain (0, ℓπ), Chen and Yu [5] chose K(x, y) = 1/ℓπ as in [11] and obtained that the constant positive steady state of model (1.2) can also lose the stability when the given parameter passes through some Hopf bifurcation values, but the bifurcating periodic solutions near such values can be spatially nonhomogeneous. This phenomenon is different from that in model (1.1) without nonlocal effect.…”