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In this paper, a mathematical model describing the interaction of malignant glioma cells, macrophages and glioma specific CD8+T cells is discussed. The biologically feasible equilibria and corresponding local stability are deduced. The bifurcations and related dynamical behaviors of this model are further studied thoroughly. The existence of transcritical bifurcation and saddle–node bifurcation is derived based on Sotomayor's theorem and Hopf bifurcation is well discussed. The codimension 2 bifurcation such as Bogdanov–Takens bifurcation is investigated using the normal form theory and center manifold theorem in more detail. Finally, numerical simulations are obtained to validate our analytical findings by varying the parameters. • The dynamics of a mathematical model describing the interaction of malignantglioma cells, macrophages and glioma specific CD8+T cells are discussed. • The co-existence of equilibria of the model are obtained by numerical simulation. • The existence of codimension 1 and codimension 2 bifurcations is investigated. • Extensive numerical simulations are obtained to validate our analytical findings. [ABSTRACT FROM AUTHOR]

In this paper, the influence of the fear effect and Leslie–Gower function on the dynamic behavior of the predator–prey model is considered. First, the well-posedness analysis of this model is demonstrated, and the existence and local stability of the equilibrium point are given. Then by using the bifurcation theory and selecting the appropriate bifurcation parameters, many types of bifurcation phenomena in the system are discovered, including transcritical bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation. Meanwhile, the numerical simulations on the basis of theoretical analysis are further carried out to intuitively show the influence of fear effect on population. As the degree of fear increases, the system will undergo multiple dynamic behaviors switching until the final extinction of prey population, while the predator population will survive due to the presence of substitute prey. In addition, the initial density of predator and prey can determine whether the solution of system tends to a coexisting steady state or a periodic oscillation. • A novel prey–predator model with the impact of fear is introduced. • The system can experience many types of bifurcation phenomena. • Numerical simulations and sensitivity analysis are performed. • Our results show that fear effect plays an important role on deriving the complex dynamics. [ABSTRACT FROM AUTHOR]

In this paper, we study the bifurcation and stability of a ratio-dependent predator-prey model with nonconstant predator harvesting rate. The analysis is carried out both analytically and numerically. We determine stability and dynamical behaviours of the equilibria of this system and characterize codimension 1 and codimension 2 bifurcations of the system analytically. Our bifurcation analysis indicates that the system exhibits numerous types of bifurcation phenomena, including Fold, Hopf, Cusp, and Bogdanov–Takens bifurcations. We use the numerical software MATCONT, to compute curves of equilibria and to compute several bifurcation curves. We especially approximate a family of limit cycles emanating from a Hopf point. Our results generalize and improve some known results and show that the model has more rich dynamics than the ratio-dependent predator-prey model without harvesting rate. [ABSTRACT FROM AUTHOR]

• Local dynamics analysis of ecological model, for host-generalist parasitoid system. • Parametric and geometric conditions to obtain Coexistence Equilibrium Points (CEPs). • Hopf and non-degenerate Bogdanov-Takens bifurcation. • Biological control of hosts can be controlled via an unstable limit cycle. • Generalist parasitoids are of fundamental importance in the stability of the host-generalist parasitoid system. Parasitoids are ubiquitous in nature and they play a fundamental role in the equilibrium of ecosystems. Their interactions with other species are particularly crucial for agricultural activities, since they act as a biological control of invasive hosts. In this paper we study Hopf, generalized Hopf and nondegenerate Bogdanov-Takens bifurcations in a simple model which describes the biological control of invasive species through predation by generalist parasitoids. We assume logistic growth for the generalist parasitoid in the absence of host species, and their interaction is described by a Holling type II functional response. We determine a geometric condition which characterizes local bifurcations associated with the appearance of periodic orbits. Further, we find conditions on the parameters which yield to the extinction of the host species, independently of the initial conditions and the charge capacity of the generalist parasitoid. Our results are supported by numerical examples. [ABSTRACT FROM AUTHOR]