A commensal symbiosis model with Holling II functional response and feedback controls is proposed and studied in this paper. The system admits four equilibria, and three boundary equilibria are unstable, only positive equilibrium is locally asymptotically stable. By applying the comparison theorem of differential equation, we show that the unique positive equilibrium is globally attractive. Numeric simulations show the feasibility of the main result.
A Lotka–Volterra commensal symbiosis model with density dependent birth rate that takes the form $$ \begin{aligned} &\frac{dx}{dt}=x \biggl( \frac{b_{11}}{b_{12}+b_{13}x}-b_{14}-a_{11}x+a_{12}y \biggr), \\ &\frac{dy}{dt}=y \biggl( \frac{b_{21}}{b_{22}+b_{23}y}-b_{24}-a_{22}y \biggr), \end{aligned} $$ where $b_{ij}$ , $i=1, 2$ , $j=1, 2, 3, 4$ , $a_{11}$ , $a_{12} $ , and $a_{22}$ are all positive constants, is proposed and studied in this paper. The system may admit four nonnegative equilibria. By constructing some suitable Lyapunov functions, we show that under some suitable assumptions, all of the four equilibria may be globally asymptotically stable, such a property is quite different to the traditional Lotka–Volterra commensalism model. With introduction of the density dependent birth rate, the dynamic behaviors of the commensalism model become complicated.
<abstract><p>A two-patch model with additive Allee effect is proposed and studied in this paper. Our objective is to investigate how dispersal and additive Allee effect have an impact on the above model's dynamical behaviours. We discuss the local and global asymptotic stability of equilibria and the existence of the saddle-node bifurcation. Complete qualitative analysis on the model demonstrates that dispersal and Allee effect may lead to persistence or extinction in both patches. Also, combining mathematical analysis with numerical simulation, we verify that the total population abundance will increase when the Allee effect constant $ a $ increases or $ m $ decreases. And the total population density increases when the dispersal rate $ D_{1} $ increases or the dispersal rate $ D_{2} $ decreases.</p></abstract>
Abstract The extinction property of a two species competitive stage-structured phytoplankton system with harvesting is studied in this paper. Several sets of sufficient conditions which ensure that one of the components will be driven to extinction are established. Our results supplement and complement the results of Li and Chen [Extinction in periodic competitive stage-structured Lotka-Volterra model with the effects of toxic substances, J. Comput. Appl. Math., 2009, 231(1), 143-153] and Liu, Chen, Luo et al. [Extinction and permanence in nonautonomous competitive system with stage structure, J. Math. Anal. Appl., 2002, 274(2), 667-684].
We investigate the stability property for the predator‐free equilibrium point of predator‐prey systems with a class of functional response and prey refuges by using the analytical approach. Under some very weakly assumption, we show that conditions that ensure the locally asymptotically stable of the predator‐free equilibrium point are consistent with that of the globally asymptotically stable ones. Our result supplements the corresponding result of Ma et al., 2009.
In this paper, we study the permanence and the periodic solution of the periodic predator-prey-mutualist system.It is well known that mutualist species can reduce the capture rate of the predator species to the prey species.By further developing the analysis technique of Teng, a set of conditions which ensure the permanence of the system are obtained.In addition, sufficient conditions are derived for the existence of positive periodic solutions to the system.An example together with its numerical simulation shows the feasibility of the main results.
Under the assumption that the feedback control variable could reduce the birth rate and increase the death rate of the species, a new type of single-species feedback control system is proposed and studied. Sufficient conditions are obtained to ensure the system admits a unique globally asymptotically stable positive equilibrium. Our research shows that if the birth rate of a species is higher than its death rate, the feedback control variable does not affect the existence and global stability of the positive equilibrium. However, as the feedback control goes up, the final density of the species goes down.
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