Asymptotic dynamics of the Leslie-Gower competition system with Allee effects and stocking

Yunshyong Chow Sophia R-J Jang 0 0 Department of Mathematics and Statistics, Texas Tech University , Lubbock, TX 79409-1042 , USA We investigate the classical Leslie-Gower competition system where one of the two competing populations is subject to Allee effects and is also under constant stocking. The model can have either no interior steady state, a unique interior steady state, two interior steady states or three interior steady states depending on parameter values. Using the tools of monotone planar systems, we provide basins of attraction for the local attractors and for the non-hyperbolic steady states. It is concluded that stocking of the weaker competitor can promote the coexistence of both competing populations. - tools of two-dimensional monotone maps [, ], we provide the asymptotic dynamics of the resulting system. The model with no stocking has been studied in []; it can have at most two interior steady states. It is proved in [] that one of the interior steady states is a saddle point and its global stable manifold is the boundary for population coexistence and competitive exclusion. With stocking, it is shown in the present study that the model can have up to three interior steady states, one of which is a saddle point and the other two are local attractors. The global stable manifold of the saddle point separates the positive cone into two positively invariant regions such that the solutions converge to either one of the interior steady states for all positive initial populations. As a consequence, stocking of one of the two competing populations with Allee effects can promote population coexistence. In the following section, we present the model and its preliminary analysis. The global behavior of the solutions is studied in Section . The final section provides discussion and conclusions. 2 The model and preliminary results Let x(t) and y(t) denote two competing populations at time t = , , . . . . The interaction between populations x and y proposed by Leslie and Gower [, ] is described by the following system: where the model has been rescaled to reduce the number of parameters. The parameters c and c are the competition coefficients and and are the intrinsic growth rates. These parameters are positive constants. The asymptotic dynamics of (.) is well studied by Cushing et al. []. Under some constraints on the parameters, the model has at most one interior steady state and the dynamics is similar to the classical Lotka-Volterra competition system of ordinary differential equations []. Kulenovi and Nurkanovi [] recently studied the following Leslie-Gower competition model with stocking: x(t + ) = y(t + ) = where h > is the constant stocking. It is proved in [] that (.) has at most two interior steady states and the basins of attractions for the steady states are explicitly provided. In [], Jang proposes the following competition system with Allee effects occurring in the x population: The expression mx+x models the mechanism of Allee effects for population x. Individuals within the population have to find others to reproduce or to exploit resources. The fraction mx+x is the probability of an individual successfully finding a mate or a cooperative individual when population size is x. The larger m the smaller this probability is, and therefore /m may be interpreted as an individual?s searching efficiency []. The resulting system can have two interior steady states in which one of them is a saddle point. The global stable manifold of the saddle point is the boundary for population coexistence and competitive exclusion []. Motivated by the work in [] and for the conservation of endangered populations, we propose the following competition system with Allee effects and stocking occurring in the x population: x(t + ) = y(t + ) = with nonnegative initial conditions, where u > denotes the constant stocking and the other parameters have the same biological meanings as those parameters in (.). If y() = , then y(t) = for t > and (.) reduces to the following scalar equation: x(t + ) = f (x) = Then f () = u, f (x) > for x > and limx f (x) = + u. An interior steady state x of (.) satisfies x + ax + ax + a = , where a = m + u , a = m u(m + ), and a = um. By the Descartes rule of signs [], the interior steady state is unique if either (a): ai > , i = , , (b): a > and a < , or (c): ai < , i = , holds. On the other hand if (d): a < and a > holds, then (.) may have three interior steady states. In the following analysis we assume that (.) has a unique steady state x, where u < x < u + . Notice f (x) < and since f is monotone increasing, we have the following asymptotic dynamics for (.). Proposition . Equation (.) has a unique steady state x, where u < x < u + and x is globally asymptotically stable in [, ) for (.). When (.) has three interior steady states, then the middle steady state is a repeller while the other two interior steady states are asymptotically stable (cf. [, Theorem .]). Similar to the proof in [], the asymptotic dynamics of (.) can be completely determined when there are three interior steady states. However, we assume (.) has a unique interior steady state for simplicity. The analysis carried out in this work can be applied to the case where (.) has three interior steady states as well. To study system (.), we first notice that (.) always has the steady state E = (x, ), where the y population is extinct. Moreover, E is the only steady state on the boundary. If , then limt y(t) = and all solutions of (.) converge to the steady state E. Therefore, we assume f(x, y) = The Jacobian matrix J of (.) evaluated at (x, y) is given by J(x, y) = f fy , y fx = ( + x+(c+yc)y(m)x + x) + ( + x +cmy)x(m + x) , det J(x, y) = ( + x +cy)x(( ++ ccxx ++ cy)y)(m + x) det J(x, y) > for x > and y . Let R+ = {(x, y) R : x , y } and let int(R+) denote the interior of R+. It is straightforward to verify that solutions of (.) remain nonnegative and are bounded for t > , and the map F is one-to-one on R . + Proposition . Solutions of (.) remain nonnegative and are bounded for t > . Moreover, the map F is one-to-one on R . + Proof Let (x, y) and (x, y) in R+ be such that F(x, y) = F(x, y), i.e., fi(x, y) = fi(x, y), i = , . If x = or y = , then (x, y) = (x, y) holds trivially. We now assume xi, yi > for i = , and x < x. Then y < y since f(x, y) = f(x, y). Let Yi = Yi(x) be defined by fi(x, Yi(x)) = fi(x, y) for i = , . Then y = Y(x) and y = Y(x) have two positive intersections (x, y) and (x, y). It is easy to see that y = Y(x) is a line with Y(x) = +ccyx > , and Y(x) > and Y (x) = (m+mx) > for x > , where = (m+x)(x+c+cy) . Moreover, a simple calculation yields Y(x) Y(x) = detfyJ(x,fyy) > by (.). Hence Y(x) > Y(x) on (x, ) and Define the partial ordering ?K ? onR+ by X = (x, y) K X = (x, y) if x x and y y, X <K X if X K X and X = X, and X K X if x < x and y > y. A map T : R+ R+ is said to be monotone with respect to (...truncated)