The effect of interference time in a predator–prey–nonprey system

In this paper, we propose a three-species model consisting of two competing (prey and nonprey) species and a predator species. Here, nonprey species are not included in the predator’s food choice. The competition process follows Holling type II competitive response to interference time. Basic results include the stability of the system. First, it is established that an increasing number of interference time stabilizes the system. Second, it is shown that the interference time has an impact on the predator equilibrium density. Third, we develop the criterion of persistence of all the species. It is also shown that the system may not be persistent when multiple steady states appear. We examine the global stability of the coexistence equilibrium point. Numerical experiments are carried out to understand the analytical outcomes.

Effect of density dependence on coinfection dynamics: part 2

In this paper we continue the stability analysis of the model for coinfection with density dependent susceptible population introduced in Andersson et al. (Effect of density dependence on coinfection dynamics. arXiv:2008.09987, 2020). We consider the remaining parameter values left out from Andersson et al. (Effect of density dependence on coinfection dynamics. arXiv:2008.09987, 2020). We look for coexistence equilibrium points, their stability and dependence on the carrying capacity K. Two sets of parameter value are determined, each giving rise to different scenarios for the equilibrium branch parametrized by K. In both scenarios the branch includes coexistence points implying that both coinfection and single infection of both diseases can exist together in a stable state. There are no simple explicit expression for these equilibrium points and we will require a more delicate analysis of these points with a new bifurcation technique adapted to such epidemic related problems. The first scenario is described by the branch of stable equilibrium points which includes a continuum of coexistence points starting at a bifurcation equilibrium point with zero single infection strain #1 and finishing at another bifurcation point with zero single infection strain #2. In the second scenario the branch also includes a section of coexistence equilibrium points with the same type of starting point but the branch stays inside the positive cone after this. The coexistence equilibrium points are stable at the start of the section. It stays stable as long as the product of K and the rate $${\bar{\gamma }}$$ γ ¯ of coinfection resulting from two single infections is small but, after this it can reach a Hopf bifurcation and periodic orbits will appear.

Stability analysis of the coexistence equilibrium of a balanced metapopulation model

We analyze the stability of a unique coexistence equilibrium point of a system of ordinary differential equations (ODE system) modelling the dynamics of a metapopulation, more specifically, a set of local populations inhabiting discrete habitat patches that are connected to one another through dispersal or migration. We assume that the inter-patch migrations are detailed balanced and that the patches are identical with intra-patch dynamics governed by a mean-field ODE system with a coexistence equilibrium. By making use of an appropriate Lyapunov function coupled with LaSalle’s invariance principle, we are able to show that the coexistence equilibrium point within each patch is locally asymptotically stable if the inter-patch dispersal network is heterogeneous, whereas it is neutrally stable in the case of a homogeneous network. These results provide a mathematical proof confirming the existing numerical simulations and broaden the range of networks for which they are valid.

How does the magnitude of genetic variation affect eco-evolutionary dynamics in character displacement?

While previous studies on character displacement tended to focus on trait divergence and convergence as a result of long-term evolution, recent studies suggest that character displacement can be a special case of evolutionary rescue, where rapid evolution prevents population extinction by weakening negative interspecific interactions. When the magnitude of genetic variation is small, however, the speed of trait divergence can be slow and populations may go extinct before the completion of character displacement. Here we analyzed a simple model to examine how the magnitude of genetic variation affects evolutionary rescue via ecological and reproductive character displacement that weakens resource competition and reproductive interference, respectively. We found that the large additive genetic variance is more important for preventing extinction in reproductive character displacement than in ecological character displacement. This is because reproductive interference produces a locally stable coexistence equilibrium with positive frequency-dependence (i.e., minority disadvantage) whereas ecological character displacement results in a globally stable coexistence equilibrium. Furthermore, population extinction becomes less likely when ecological and reproductive character displacement occur simultaneously due to positive covariance between ecological and reproductive traits. Our results suggest that while reproductive character displacement may be rarer than ecological character displacement, it is more likely to occur when there exists positive trait covariance, such as the case of a magic trait in reinforcement of speciation processes.

Qualitative properties of mathematical model of English language education

This paper presents a mathematical model that examines the impacts of traditional and modern educational programs. We calculate two reproduction numbers. By using the Chavez and Song theorem, we show that backward bifurcation occurs. In addition, we investigate the existence and local and global stability of boundary equilibria and coexistence equilibrium point and the global stability of the coexistence equilibrium point using compound matrices.

Qualitative Analysis of the Effect of Weeds Removal in Paddy Ecosystems in Fallow Season

In the paper, we introduce a differential equations model of paddy ecosystems in the fallow season to study the effect of weeds removal from the paddy fields. We found that there is an unstable equilibrium of the extinction of weeds and herbivores in the system. When the intensity of weeds removal meets certain conditions and the intrinsic growth rate of herbivores is higher than their excretion rate, there is a coexistence equilibrium state in the system. By linearizing the system and using the Routh–Hurwitz criterion, we obtained the local asymptotically stable conditions of the coexistence equilibrium state. The critical value formula of the Hopf bifurcation is presented too. The model demonstrates that weeds removal from paddy fields could largely reduce the weeds biomass in the equilibrium state, but it also decreases the herbivore biomass, which probably reduces the content of inorganic fertilizer in the soil. We found a particular intensity of weeds removal that could result in the minimum content of inorganic fertilizer, suggesting weeds removal should be kept away from this intensity.

Impact of the Fear Effect on the Stability and Bifurcation of a Leslie–Gower Predator–Prey Model

In this paper, we investigate analytically and numerically the dynamics of a modified Leslie–Gower predator–prey model which is characterized by the reduction of prey growth rate due to the anti-predator behavior. We prove the existence and local/global stability of equilibria of the model, and verify the existence of Hopf bifurcation. In addition, we focus on the influence of the fear effect on the population dynamics of the model and find that the fear effect can not only reduce the population density of both predator and prey, but also destabilize the coexistence equilibrium, which are beneficial to the occurrence of limit-cycle-induced oscillation, or prevent the occurrence of limit cycle oscillation and increase the stability of the system.

Complex dynamics of a fractional-order predator-prey interaction with harvesting

Harvesting has a strong impact on the dynamic evolution of a population subjected to it. In this paper, a fractional-order predator-prey interaction is studied with harvesting affecting both predator and prey populations. Local stability of the coexistence equilibrium point is discussed depending upon the harvesting of prey. Moreover, period-doubling and Neimark-Sacker bifurcations are studied for a wide range of constant harvesting effort of prey.