Non-existence of non-constant positive steady states of two Holling type-II predator–prey systems: Strong interaction case

Abstract

We prove the non-existence of non-constant positive steady state solutions of two reaction-diffusion predator-prey models with Holling type-II functional response when the interaction between the predator and the prey is strong. The result implies that the global bifurcating branches of steady state solutions are bounded loops.

Existence and non-existence of spatial patterns in a ratio-dependent predator–prey model

“…These analytical results are unable to predict the generation of spatial patterns due to large spatiotemporal perturbation from the homogeneous steady-state. There exist some analytical techniques to derive mathematical criteria for the stability of homogeneous steady-state and derivation of appropriate conditions for the existence and/or non-existence of non-constant stationary states (Pang and Wang, 2003;Peng and Shi, 2009;Peng et al, 2008;Shi, 2002;Shi et al, 2010;Smoller, 1994;Wang et al, 2011). These techniques are already utilized by the researchers to obtain analytical results for some spatiotemporal models of interacting populations but those results are not cross verified with the scenario of spatial pattern formation for specific choices of the parameters involved with the targeted models.…”

Coexistence states of a Holling type-II predator–prey system

“…We remark that problem (1.1) with homogeneous Neumann boundary condition was discussed in [25,30] recently. For simplicity, we make the following transformation…”

“…In the spatially inhomogeneous case, the existence of a noconstant time-independent positive solution, also called stationary pattern, is an indication of the richness of the corresponding partial differential equation dynamics. In recent years, stationary pattern induced by diffusion has been studied extensively, and many important phenomena have been observed (see [4,[9][10][11]13,14,[20][21][22][23][24][25][26]28,31] and references therein).…”

Positive periodic solutions of neutral predator–prey model with Beddington–DeAngelis functional response

“…There is a large body of literature on population dynamics in ecological modelling, particularly in predator-prey systems [1][2][3]. A well-known model of such systems is the predator-prey model with a Beddington-DeAngelis functional response.…”