Bifurcation analysis in a diffusive predator–prey system with Michaelis–Menten-type predator harvesting

Abstract

In this paper, we consider a modified predator-prey model with Michaelis-Menten-type predator harvesting and diffusion term. We give sufficient conditions to ensure that the coexisting equilibrium is asymptotically stable by analyzing the distribution of characteristic roots. We also study the Turing instability of the coexisting equilibrium. In addition, we use the natural growth rate r 1 of the prey as a parameter and carry on Hopf bifurcation analysis including the existence of Hopf bifurcation, bifurcation direction, and the stability of the bifurcating periodic solution by the theory of normal form and center manifold method. Our results suggest that the diffusion term is important for the study of the predator-prey model, since it can induce Turing instability and spatially inhomogeneous periodic solutions. The natural growth rate r 1 of the prey can also affect the stability of positive equilibrium and induce Hopf bifurcation.

Stability and bifurcation analysis in a single-species stage structure system with Michaelis–Menten-type harvesting

“…With more works on the harvesting model published, some scholars began to focus their attention to the so called Michaelis-Menten-type harvesting [27][28][29][30][31][32][33]. The harvesting term takes the form of H = hEx mE+nx , which was first proposed by Clark [34].…”

Analysis of Structure-Preserving Discrete Models for Predator-Prey Systems with Anomalous Diffusion

“…Indeed, this area of research has proven to be extremely fruitful in view of the wide range of possible scenarios that merit investigation. As examples, we can mention studies that report on the modeling and analysis of predator-prey models with disease in the prey [1], the analysis of stochastic systems with modified Leslie-Gower and Holling-type schemes [2], the dynamic behaviors of Lotka-Volterra predator-prey models that incorporate predator cannibalism [3], the analysis of diffusive predator-prey systems with Michaelis-Menten-type predator harvesting [4], synthetic Escherichia coli predator-prey ecosystems [5], the analytical investigation of stage-structured predator-prey models depending on maturation delay and death rate [6], and non-autonomous ratio dependent models with Holling-type functional response with temporal delay [7], among other interesting topics [8].…”

Stability and bifurcation analysis in a delay-induced predator-prey model with Michaelis-Menten type predator harvesting

“…Effects of harvesting in various types of prey-predator models have been considered by many researchers [7,15,33,34,39,47]. It has shown that harvesting has a strong influence on the dynamical behavior of a predator-prey model [1,19,21,37,42]. Such as appearance of numerous kinds of bifurcations, including saddle-node bifurcation, Hopf bifurcation, repelling and attracting Bogdanov-Takens bifurcations of codimensions 2 and 3.…”