Dynamics in a diffusive predator–prey system with double Allee effect and modified Leslie–Gower scheme

In this paper, we investigate a diffusive modified Leslie–Gower predator–prey system with double Allee effect on prey. The global existence, uniqueness and a priori bound of positive solutions are determined. The existence and local stability of constant steady–state solutions are analyzed. Next, we induce the nonexistence of nonconstant positive steady–state solutions, which indicates the effect of large diffusivity. Furthermore, we discuss the steady–state bifurcation and the existence of nonconstant positive steady–state solutions by the bifurcation theory. In addition, Hopf bifurcations of the spatially homogeneous and inhomogeneous periodic orbits are studied. Finally, we make some numerical simulations to validate and complement the theoretical analysis. Our results demonstrate that the dynamics of the system with double Allee effect and modified Leslie–Gower scheme are richer and more complex.

Steady-State Problem of an S-K-T Competition Model with Spatially Degenerate Coefficients

In this paper, we study the steady-state problem of an S-K-T competition model with a spatially degenerate intraspecific competition coefficient. First, the global bifurcation continuum of positive steady-state solutions from its semitrivial steady-state solution is given, which depends on the spatial heterogeneity and cross-diffusion. Second, two limiting systems are derived as the cross-diffusion coefficient tends to infinity. Moreover, we demonstrate the existence of positive steady-state solutions near the two limiting systems, and show which one of the limiting systems characterizes the positive steady-state solution.

Global Bifurcation Structure of a Predator-Prey System with a Spatial Degeneracy and B-D Functional Response

In this paper, we investigate a predator-prey system with Beddington–DeAngelis (B-D) functional response in a spatially degenerate heterogeneous environment. First, for the case of the weak growth rate on the prey ( λ 1 Ω < a < λ 1 Ω 0 ), a priori estimates on any positive steady-state solutions are established by the comparison principle; two local bifurcation solution branches depending on the bifurcation parameter are obtained by local bifurcation theory. Moreover, the demonstrated two local bifurcation solution branches can be extended to a bounded global bifurcation curve by the global bifurcation theory. Second, for the case of the strong growth rate on the prey ( a > λ 1 Ω 0 ), a priori estimates on any positive steady-state solutions are obtained by applying reduction to absurdity and the set of positive steady-state solutions forms an unbounded global bifurcation curve by the global bifurcation theory. In the end, discussions on the difference of the solution properties between the traditional predator-prey system and the predator-prey system with a spatial degeneracy and B-D functional response are addressed.

Bifurcation analysis in a diffusive phytoplankton–zooplankton model with harvesting

A diffusive phytoplankton–zooplankton model with nonlinear harvesting is considered in this paper. Firstly, using the harvesting as the parameter, we get the existence and stability of the positive steady state, and also investigate the existence of spatially homogeneous and inhomogeneous periodic solutions. Then, by applying the normal form theory and center manifold theorem, we give the stability and direction of Hopf bifurcation from the positive steady state. In addition, we also prove the existence of the Bogdanov–Takens bifurcation. These results reveal that the harvesting and diffusion really affect the spatiotemporal complexity of the system. Finally, numerical simulations are also given to support our theoretical analysis.

Dynamical analysis of a reaction–diffusion SEI epidemic model with nonlinear incidence rate

In this paper, a reaction–diffusion SEI epidemic model with nonlinear incidence rate is proposed. The well-posedness of solutions is studied, including the existence of positive and unique classical solution and the existence and the ultimate boundedness of global solutions. The basic reproduction numbers are given in both heterogeneous and homogeneous environments. For spatially heterogeneous environment, by the comparison principle of the diffusion system, the infection-free steady state is proved to be globally asymptotically stable if [Formula: see text] if [Formula: see text], the system will be persistent and admit at least one positive steady state. For spatially homogenous environment, by constructing a Lyapunov function, the infection-free steady state is proved to be globally asymptotically stable if [Formula: see text] and then the unique positive steady state is achieved and is proved to be globally asymptotically stable if [Formula: see text]. Finally, two examples are given via numerical simulations, and then some control strategies are also presented by the sensitive analysis.

Stability of positive steady-state solutions to a time-delayed system with some applications

<p style='text-indent:20px;'>In this paper, we study a general nonlinear retarded system:</p><p style='text-indent:20px;'><disp-formula> <label>1</label> <tex-math id="E1"> \begin{document}$ \begin{equation} y'(t) = a(t)F(y(t),y(t-\tau)), \; \; t\geq 0, \end{equation} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \tau&gt;0 $\end{document}</tex-math></inline-formula> is a constant, <inline-formula><tex-math id="M2">\begin{document}$ a(t) $\end{document}</tex-math></inline-formula> is a positive value function defined on <inline-formula><tex-math id="M3">\begin{document}$ [0,\infty) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ F(y,z) $\end{document}</tex-math></inline-formula> is continuous in <inline-formula><tex-math id="M5">\begin{document}$ \mathscr{D} = \mathbb{R}_+^2 $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M6">\begin{document}$ \mathbb{R_+} = (0,+\infty) $\end{document}</tex-math></inline-formula>. Sufficient conditions for stability of the unique positive equilibrium are established. Our results show that if <inline-formula><tex-math id="M7">\begin{document}$ F_z(y,z)&gt;0 $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M8">\begin{document}$ y,z\in \mathbb{R_+} $\end{document}</tex-math></inline-formula>, then the unique positive equilibrium of (1) which denoted by <inline-formula><tex-math id="M9">\begin{document}$ \bar{y} $\end{document}</tex-math></inline-formula> is globally stable for any positive initial value and all <inline-formula><tex-math id="M10">\begin{document}$ \tau&gt;0 $\end{document}</tex-math></inline-formula>; if <inline-formula><tex-math id="M11">\begin{document}$ F(y,z) $\end{document}</tex-math></inline-formula> is decreasing in <inline-formula><tex-math id="M12">\begin{document}$ y $\end{document}</tex-math></inline-formula>, then <inline-formula><tex-math id="M13">\begin{document}$ \bar{y} $\end{document}</tex-math></inline-formula> is globally stable for small <inline-formula><tex-math id="M14">\begin{document}$ \tau $\end{document}</tex-math></inline-formula>. Some applications are given.</p>

Analysis of a diffusive host-pathogen model with standard incidence and distinct dispersal rates

This paper concerns with detailed analysis of a reaction-diffusion host-pathogen model with space-dependent parameters in a bounded domain. By considering the fact the mobility of host individuals playing a crucial role in disease transmission, we formulate the model by a system of degenerate reaction-diffusion equations, where host individuals disperse at distinct rates and the mobility of pathogen is ignored in the environment.We first establish the well-posedness of the model, including the global existence of solution and the existence of the global compact attractor. The basic reproduction number is identified, and also characterized by some equivalent principal spectral conditions, which establishes the threshold dynamical result for pathogen extinction and persistence. When the positive steady state is confirmed, we investigate the asymptotic profiles of positive steady state as host individuals disperse at small and large rates. Our result suggests that small and large diffusion rate of hosts have a great impacts in formulating the spatial distribution of the pathogen.

A hidden integral structure endows absolute concentration robust systems with resilience to dynamical concentration disturbances

Biochemical systems that express certain chemical species of interest at the same level at any positive steady state are called ‘absolute concentration robust’ (ACR). These species behave in a stable, predictable way, in the sense that their expression is robust with respect to sudden changes in the species concentration, provided that the system reaches a (potentially new) positive steady state. Such a property has been proven to be of importance in certain gene regulatory networks and signaling systems. In the present paper, we mathematically prove that a well-known class of ACR systems studied by Shinar and Feinberg in 2010 hides an internal integral structure. This structure confers these systems with a higher degree of robustness than was previously known. In particular, disturbances much more general than sudden changes in the species concentrations can be rejected, and robust perfect adaptation is achieved. Significantly, we show that these properties are maintained when the system is interconnected with other chemical reaction networks. This key feature enables the design of insulator devices that are able to buffer the loading effect from downstream systems—a crucial requirement for modular circuit design in synthetic biology. We further note that while the best performance of the insulators are achieved when these act at a faster timescale than the upstream module (as typically required), it is not necessary for them to act on a faster timescale than the downstream module in our construction.