Mathematical analysis of toxin-phytoplankton-fish model with self-diffusion and cross-diffusion

Abstract

In this paper we propose a nonlinear reaction-diffusion system describing the interaction between toxin-producing phytoplankton and fish population. We analyze the effect of self- and cross-diffusion on the dynamics of the system. The existence, uniqueness and uniform boundedness of solutions are established in the positive octant. The system is analyzed for various interesting dynamical behaviors which include boundedness, persistence, local stability, global stability around each equilibria based on some conditions on self- and cross-diffusion coefficients. The analytical findings are verified by numerical simulation.

A mathematical analysis of Hopf-bifurcation in a prey-predator model with nonlinear functional response

“…Remark 2 The biological interpretation of the Hopf-bifurcation is that the prey coexists with the predator, exhibiting oscillatory equilibrium behavior [10,11]. Indeed, we observe that if the predation threshold δ 1 > δ 1c , we have periodic fluctuation of the prey and predator species: Figures 5(c) and 5(d) show the existence of a limit cycle resulting from the Hopf-bifurcation.…”

“…In this subsection, we define the conditions of Hopf-bifurcations and the critical values of Hopf bifurcations. Here, δ 1 is taken as a bifurcation parameter [10,15,31]. .…”

“…These different types of functional responses present a key element for understanding the dynamics of these populations. The main questions concerning population dynamics concern the effects of interaction in the regulation of natural populations, the reduction of their size, the reduction of their natural fluctuations, or the destabilization of the equilibria in oscillations of the states of the population [8][9][10][11][12][13]. The predator-prey relationship is important to maintain the balance between different animal species.…”