The generalised Lotka-Volterra system describing N interacting species of densities X = (X1, X2,...,xn)" is given by the

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The Generalised Lotka Volterra System Describing N Interacting Species Of Densities X X1 X2 Xn Is Given By The 1 (41.67 KiB) Viewed 21 times

The generalised Lotka-Volterra system describing N interacting species of densities X = (X1, X2,...,xn)" is given by the equations: x =x(n+Šax) Xi X X; i=1,...,N. Here diy are the elements of the interaction matrix A and r = (1, 12,... ,")is the intrinsic growth vector. Each component of r can be positive, negative or zero. (a) State Lypanunov's theorem for the asymptotic stability of a steady state. (b) Prove that if the system has a steady state X inside the positive orthant (R+N), and the matrix A is negative definite, then the steady state is globally attracting inside the positive orthant. Hint: Consider the function V(X)-(x), where 6(E) ( 6-x; – x; los(19)) X (c) Given the following interaction matrix for a system composed of 3 species -30 2 A= 0 -31 1 1 -1 C ) discuss carefully the interactions between the species and show that this interaction matrix does not support a unique interior steady state. (d) For the interaction matrix given in part (c), by considering log(71723) or otherwise, show that interior orbits are unbounded if rı +2+3r3 > 0.