Modeling the Phenomenon of Xenophobia in Africa

In this study, we applied the principle of a competitive predator-prey system to propose a prey-predator-like model of xenophobia in Africa. The boundedness of the solution, the existence and stability of equilibrium states of the xenophobic model are discussed accordingly. As a special case, the coexistence state was found to be locally and globally stable based on the parametric conditions of effective group defense and anti-xenophobic policy implementation. The system was further analyzed by Sotomayor’s theory to show that each equilibrium point bifurcates transcritically. However, numerical proof showed period-doubling bifurcation, which makes the xenophobic situation more chaotic in Africa. Further numerical simulations support the analytical results with the view that tolerance, group defense and anti-xenophobic policies are critical parameters for the coexistence of foreigners and xenophobes.

Dynamics of a discrete-time pioneer–climax model

We present a simple mathematical model for the dynamics of a successional pioneer–climax system using difference equations. Each population is subject to inter- and intraspecific competition; population growth is dependent on the combined densities of both species. Nine different geometric cases, corresponding to different orientations of the zero-growth isoclines, are possible for this system. We fully characterize the long-term dynamics of the model for each of the nine cases, uncovering diverse sets of potential behaviors. Competitive exclusion of the pioneer species and of the climax species are both possible depending on the relative strength of competition. Stable coexistence of both species may also occur; in two cases, a coexistence state is destabilized through a Neimark–Sacker bifurcation, and an attracting invariant circle is born. The invariant circle eventually disappears into thin air in a heteroclinic or homoclinic bifurcation, leading to the sudden transition of the system to an exclusion state. Neither global bifurcation has been observed in a discrete-time pioneer–climax model before. The homoclinic bifurcation is novel to all pioneer–climax models. We conclude by discussing the ecological implications of our results.

Stochasticity-induced stabilization weakens in diverse communities.

Environmental stochasticity and the temporal variations of demographic rates associated with it are ubiquitous in nature. The ability of these fluctuations to stabilize a coexistence state of competing populations (sometimes known as the storage effect) is a counterintuitive feature that has aroused much interest. Here we consider the performance of environmental stochasticity as a stabilizer in diverse communities. We show that the effect of stochasticity is buffered because of the differential response of populations to environmental variations, and its stabilizing effect disappears as the number of populations increases. Of particular importance is the ratio between the autocorrelation time of the environment and the generation time. Species richness grows with stochasticity only when this ratio is smaller than the inverse of the fundamental biodiversity parameter. When stochasticity impedes coexistence and lowers the species richness, the ratio between the strength of environmental variations and the speciation (or migration) rate governs its effect.

Dynamics of a Discrete-time Pioneer–climax Model

We present a simple mathematical model for the dynamics of a successional pioneer–climax system using difference equations. Each population is subject to inter- and intraspecific competition; population density growth is dependent on the combined densities of both species. Nine different geometric cases, corresponding to different orientations of the zero-growth isoclines, are possible for this system. We fully characterize the long-term dynamics of the model for each of the nine cases, uncovering diverse sets of potential behaviors. Competitive exclusion of the pioneer species and exclusion of the climax species are both possible depending on the relative strength of competition. Stable coexistence of both species may also occur; in two cases, a coexistence state is destabilized through a Neimark–Sacker bifurcation and an attracting invariant circle is born. The invariant circle eventually disappears into thin air in a heteroclinic or homoclinic bifurcation, leading to the sudden transition of the system to an exclusion state. Neither global bifurcation has been observed in a discrete-time pioneer–climax model before. The homoclinic bifurcation is novel to all pioneer–climax models. We conclude by discussing the ecological implications of our results.

Positive solutions of a diffusive two competitive species model with saturation

<p style='text-indent:20px;'>In this paper, the positive solutions of a diffusive two competitive species model with Bazykin functional response are investigated. We give the a priori estimates and compute the fixed point indices of trivial and semi-trivial solutions. And obtain the existence of solution and demonstrate the bifurcation of a coexistence state emanating from semi-trivial solutions. Finally, multiplicity and stability results are presented.</p>

Large time behavior in a predator-prey system with pursuit-evasion interaction

<p style='text-indent:20px;'>This work considers a pursuit-evasion model</p><p style='text-indent:20px;'><disp-formula><label/><tex-math id="FE1000">\begin{document}$\begin{equation} \left\{ \begin{split} &amp;u_t = \Delta u-\chi\nabla\cdot(u\nabla w)+u(\mu-u+av),\\ &amp;v_t = \Delta v+\xi\nabla\cdot(v\nabla z)+v(\lambda-v-bu),\\ &amp;w_t = \Delta w-w+v,\\ &amp;z_t = \Delta z-z+u\\ \end{split} \right. \ \ \ \ \ (1) \end{equation}$\end{document}</tex-math></disp-formula></p><p style='text-indent:20px;'>with positive parameters <inline-formula><tex-math id="M1">\begin{document}$ \chi $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ \xi $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ \mu $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ a $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ b $\end{document}</tex-math></inline-formula> in a bounded domain <inline-formula><tex-math id="M7">\begin{document}$ \Omega\subset\mathbb{R}^N $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M8">\begin{document}$ N $\end{document}</tex-math></inline-formula> is the dimension of the space) with smooth boundary. We prove that if <inline-formula><tex-math id="M9">\begin{document}$ a&lt;2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M10">\begin{document}$ \frac{N(2-a)}{2(C_{\frac{N}{2}+1})^{\frac{1}{\frac{N}{2}+1}}(N-2)_+}&gt;\max\{\chi,\xi\} $\end{document}</tex-math></inline-formula>, (1) possesses a global bounded classical solution with a positive constant <inline-formula><tex-math id="M11">\begin{document}$ C_{\frac{N}{2}+1} $\end{document}</tex-math></inline-formula> corresponding to the maximal Sobolev regularity. Moreover, it is shown that if <inline-formula><tex-math id="M12">\begin{document}$ b\mu&lt;\lambda $\end{document}</tex-math></inline-formula>, the solution (<inline-formula><tex-math id="M13">\begin{document}$ u,v,w,z $\end{document}</tex-math></inline-formula>) converges to a spatially homogeneous coexistence state with respect to the norm in <inline-formula><tex-math id="M14">\begin{document}$ L^\infty(\Omega) $\end{document}</tex-math></inline-formula> in the large time limit under some exact smallness conditions on <inline-formula><tex-math id="M15">\begin{document}$ \chi $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M16">\begin{document}$ \xi $\end{document}</tex-math></inline-formula>. If <inline-formula><tex-math id="M17">\begin{document}$ b\mu&gt;\lambda $\end{document}</tex-math></inline-formula>, the solution converges to (<inline-formula><tex-math id="M18">\begin{document}$ \mu,0,0,\mu $\end{document}</tex-math></inline-formula>) with respect to the norm in <inline-formula><tex-math id="M19">\begin{document}$ L^\infty(\Omega) $\end{document}</tex-math></inline-formula> as <inline-formula><tex-math id="M20">\begin{document}$ t\rightarrow \infty $\end{document}</tex-math></inline-formula> under some smallness assumption on <inline-formula><tex-math id="M21">\begin{document}$ \chi $\end{document}</tex-math></inline-formula> with arbitrary <inline-formula><tex-math id="M22">\begin{document}$ \xi $\end{document}</tex-math></inline-formula>.</p>

Individual Variability in Dispersal and Invasion Speed

We model the growth, dispersal and mutation of two phenotypes of a species using reaction–diffusion equations, focusing on the biologically realistic case of small mutation rates. Having verified that the addition of a small linear mutation term to a Lotka–Volterra system limits it to only two steady states in the case of weak competition, an unstable extinction state and a stable coexistence state, we exploit the fact that the spreading speed of the system is known to be linearly determinate to show that the spreading speed is a nonincreasing function of the mutation rate, so that greater mixing between phenotypes leads to slower propagation. We also find the ratio at which the phenotypes occur at the leading edge in the limit of vanishing mutation.

Biological control of a predator–prey system through provision of an infected predator

Epidemic transmission has a substantial effect on the dynamics and stability of the predator–prey system, in which the transmission rate plays an important role. The probabilistic cellular automaton (PCA) approach is used to investigate the spatiotemporal dynamics of a predator–prey system with the infected predator. Remarkably, it is impossible to achieve a coexistence state of prey, susceptible predators, and infected predators in a spatial population. This is different from the analysis from a non-spatial population with the mean-field approximation, where Hopf bifurcation arises and the interior equilibrium becomes unstable, and a periodic solution appears with the increasing infection rate. The results show that the introduction of the infected predator with a high transmission rate is beneficial for the persistence of the prey population in space. However, a low transmission rate will promote the coexistence state of the prey and the susceptible predator populations. In summary, it is possible to develop management strategies to manipulate the transmission rate of the infected predator for the benefit of biological control.