Turing–Hopf Bifurcation Analysis for a Diffusive Prey–Predator Model with a Chemotactic Term

Since the ratio-dependent theory reflects the fact that predators must share and compete for food, it is suitable for describing the relationship between predators and their preys and has recently become a very important theory put forward by biologists. In order to investigate the dynamical relationship between predators and their preys, a so-called Michaelis-Menten ratio-dependent predator-prey model is studied in this paper with gestation time delays of predators and preys taken into consideration. The stability of the positive equilibrium is investigated by the Nyquist criteria, and the existence of the local Hopf bifurcation is analyzed by employing the theory of Hopf bifurcation. By means of the center manifold and the normal form theories, explicit formulae are derived to determine the stability, direction and other properties of bifurcating periodic solutions. The above theoretical results are validated by numerical simulations with the help of dynamical software WinPP. The results show that if both the gestation delays are small enough, their sizes will keep stable in the long run, but if the gestation delays of predators are big enough, their sizes will periodically fluctuate in the long term. In order to reveal the effects of time delays on the ratio-dependent predator-prey model, a ratio-dependent predator-prey model without time delays is considered. By Hurwitz criteria, the local stability of positive equilibrium of this model is investigated. The conditions under which the positive equilibrium is locally asymptotically stable are obtained. By comparing the results with those of the model with time delays, it shows that the dynamical behaviors of ratio-dependent predator-prey model with time delays are more complicated. Under the same conditions, namely, with the same parameters, the stability of positive equilibrium of ratio-dependent predator-prey model would change due to the introduction of gestation time delays for predators and preys. Moreover, with the variation of time delays, the positive equilibrium of the ratio-dependent predator-prey model subjects to Hopf bifurcation.

In this article, we study Hopf bifurcation and Turing instability of a diffusive predator-prey model with hunting cooperation. For the local model, we analyze the stability of the equilibrium and derive conditions for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solution by the center manifold and the normal form theory. For the reaction-diffusion model, first it is shown that Turing instability occurs, then the direction and stability of the Hopf bifurcation is reached. Our results show that hunting cooperation plays a crucial role in the dynamics of the model, that is, it can be beneficial to the predator population and induce the rise of Turing instability. Finally, numerical simulations are performed to visualize the complex dynamic behavior.

In this paper, we investigate the effect of nonlocal delay on stability and spatiotemporal dynamics of a diffusive predator-prey model with Allee effect in predators. For the system without nonlocal delay, the stability and Turing instability conditions are derived. For the system with nonlocal delay, all Turing bifurcation curves on $ (d_{2}, \tau) $ plane are obtained, and the critical Turing bifurcation curve near the stable region is determined by the discussion of the properties of Turing bifurcation curves. It is found that the presence of nonlocal delay decreases the range of the occurrence for Turing bifurcation. The critical Hopf bifurcation curve is found by the geometric stability switch criterion. The stable region for the positive equilibrium is bounded by the critical Turing and Hopf bifurcation curves. On the boundaries of stable region, Hopf bifurcation, Turing bifurcation, Turing-Turing bifurcation and Turing-Hopf bifurcation may occur. To reveal the complex dynamic near Turing-Hopf bifurcation point, normal form for the system with nonlocal delay is derived, from which different types of bistability are found, including two stable spatially inhomogeneous periodic solutions and two stable spatially inhomogeneous steady states.

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.

In this paper, the dynamic behaviors of coupled reaction–diffusion neural oscillator system with excitatory-to-inhibitory connection and time delay under the Neumann boundary conditions are investigated. By constructing a basis of phase space based on the eigenvectors of Laplace operator, the characteristic equation of this system is obtained. Then, the local stability of zero solution and the occurrence of Hopf bifurcation are established by regarding the time delay as the bifurcation parameter. In particular, by using the normal form theory and center manifold theorem of the partial differential equation, the normal forms are obtained, which determine the bifurcation direction and the stability of the periodic solutions. Finally, two examples are given to verify the theoretical results.

In this paper, we present an algorithm for deriving the normal forms of Bautin bifurcations in reaction-diffusion systems with time delays and Neumann boundary conditions. On the center manifold near a Bautin bifurcation, the first and second Lyapunov coefficients are calculated explicitly, which completely determine the dynamical behavior near the bifurcation point. As an example, the Segel-Jackson predator-prey model is studied. Near the Bautin bifurcation we find the existence of fold bifurcation of periodic orbits, as well as subcritical and supercritical Hopf bifurcations. Both theoretical and numerical results indicate that solutions with small (large) initial conditions converge to stable periodic orbits (diverge to infinity).

In this paper, a two-delay HIV-1 virus model with delay-dependent parameters is considered. The model includes both virus-to-cell and cell-to-cell transmissions. Firstly, immune-inactivated reproduction rate R 0 and immune-activated reproduction rate R 1 are deduced. When R 1 > 1 , the system has the unique positive equilibrium E ∗ . The local stability of the positive equilibrium and the existence of Hopf bifurcation are obtained by analyzing the characteristic equation at the positive equilibrium with the time delay as the bifurcation parameter and four different cases. Besides, we obtain the direction and stability of the Hopf bifurcation by using the center manifold theorem and the normal form theory. Finally, the theoretical results are validated by numerical simulation.

Analysis of stability and Hopf bifurcation for a viral infectious model with delay

In this paper, the dynamics of a spruce-budworm model with delay is investigated. We show that there exists Hopf bifurcation at the positive equilibrium as the delay increases. Some sufficient conditions for the existence of Hopf bifurcation are obtained by investigating the associated characteristic equation. By using the theory of normal form and center manifold, explicit expression for determining the direction of Hopf bifurcations and the stability of bifurcating periodic solutions are presented.

Hopf bifurcation in a reaction-diffusion system governed by the fitz-hugh-nagumo equations

Stability and bifurcation analysis on a Logistic model with discrete and distributed delays

This paper presents the analysis of the global bifurcations and chaotic dynamics for the nonlinear nonplanar oscillations of a cantilever beam subjected to a harmonic axial excitation and transverse excitations at the free end. The governing nonlinear equations of nonplanar motion with parametric and external excitations are obtained. The Galerkin procedure is applied to the partial differential governing equation to obtain a two-degree-of-freedom nonlinear system with parametric and forcing excitations. The resonant case considered here is 2:1 internal resonance, principal parametric resonance-1/2 subharmonic resonance for the in-plane mode and fundamental parametric resonance–primary resonance for the out-of-plane mode. The parametrically and externally excited system is transformed to the averaged equations by using the method of multiple scales. From the averaged equation obtained here, the theory of normal form is applied to find the explicit formulas of normal forms associated with a double zero and a pair of pure imaginary eigenvalues. Based on the normal form obtained above, a global perturbation method is utilized to analyze the global bifurcations and chaotic dynamics in the nonlinear nonplanar oscillations of the cantilever beam. The global bifurcation analysis indicates that there exist the heteroclinic bifurcations and the Silnikov type single-pulse homoclinic orbit in the averaged equation for the nonlinear nonplanar oscillations of the cantilever beam. These results show that the chaotic motions can occur in the nonlinear nonplanar oscillations of the cantilever beam. Numerical simulations verify the analytical predictions.

In this paper, we deal with the Gause–Kolmogorov-type predator–prey system with indirect prey-taxis, which means that directional movement of predators is stimulated by some chemicals emitted by preys. The existence of the positive equilibrium, the effect of the indirect prey-taxis on the stability and the related bifurcations are investigated. The critical values for the occurrence of the Hopf bifurcation, Turing bifurcation, Turing–Hopf bifurcation and double-Hopf bifurcation are explicitly determined. An algorithm for calculating the normal form of the double-Hopf bifurcation for the non-resonance and weak resonance is derived. Moreover, we apply the theoretical results to the system with Holling-II type functional response, the stable region and the bifurcation curves are completely determined in the plane of the indirect prey-taxis and self saturation coefficient. The dynamical classification near the double-Hopf bifurcation point is explicitly determined. In the neighborhood of the double-Hopf bifurcation, there are stable spatially homogeneous/inhomogeneous periodic solutions, stable spatially inhomogeneous quadi-periodic solutions and the pattern transitions from one spatial–temporal patterns to another one with the changes of the indirect taxis and semi saturation coefficients. The results show that spatially inhomogeneous Hopf bifurcations are induced by an indirect prey-taxis parameter $$\chi >0$$ , which is impossible for the reaction–diffusion predator–prey model with a direct prey-taxis.

Spatiotemporal dynamics in a diffusive predator-prey model with multiple Allee effect and herd behavior

Characterization of stability limits of Ledinegg instability and density wave oscillations for two-phase flow in natural circulation loops