Global dynamics of a cooperative system with ceiling density dependence

Document Type

Article

Date of Original Version

1-1-2019

Abstract

We investigate the global behavior of the cooperative system xt+1 = minfr11xt + r12yt;K1g; yt+1 = minfr21xt + r22yt;K2g; t = 0, 1.... where the initial conditions x0, y0 are arbitrary nonnegative numbers. This system models a population comprised of two subpopulations on different patches of land. The model, introduced in [4], considers the minimum between the maximum carrying capacity of each patch (K1or K2resp.) and the linear combination of the population from patch i from the last time step with those who migrated to patch i for i=1,2. We break the behavior of the system into several cases based on whether the linear combination of the population or maximum carrying capacity is greater. We are able to conclude that either one fixed point will be a global attractor of the interior region of R2+or there will exist a line of fixed points with the stable manifolds as the basins of attractions. We then extend some of these results to the n-dimensional case, first introduced in [2] using similar techniques. We investigate the global behavior of general cooperative system xit+1= min{ri1x1t+ ri2x2t+ .... + riixit+ .... + rinxnt} for i = 1, 2...., n, and t=0,1.... where the initial conditions of xi 0 are arbitrary nonnegative numbers for i=1,2, : : :, n. We are able to conclude that in some cases one fixed point will be a global attractor of the interior region of Rn+

Publication Title, e.g., Journal

International Journal of Difference Equations

Volume

14

Issue

1

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