Nonautonomous logistic equations as models of the adjustment of populations to environmental change

Abstract

For each choice r and K as functions mapping (−∞, ∞) into a compact subset of (0, ∞), the nonautonomous logistic equation, x ̇ (t)=r(t)x(t) 1− x(t) K(t) , (a) possesses a special solution x ∗:(−∞, ∞)→(0, ∞), here called the canonical solution, which is approached, in the limit of large t, by each solution obeying an initial condition of the form x( t 0) = x 0> 0. At each time t, the value x ∗( t) of x ∗ is given by a functional F of the histories of r and K up to t, i.e., x ∗(t)= F(r t,K t) , (b) where r t ( s)= r( t− s) and K t ( s)= K( t− s) for all s⩾0. The functional F , although nonlinear has a simple form and possesses regularity properties of the type assumed in the theory of “fading memory”. Results from that theory are applied to obtain asymptotic approximations to F appropriate for those cases in which the “carrying capacity of the environment,” K, varies slowly in time and for those cases in which K fluctuates at an arbitrary rate, but remains close to a constant value. For each choice of r, K, t 0 , and x 0 , the solution x of (a) obeying the initial condition x( t 0)= x 0>0, is given by x(t)= F(r t,T (t−t 0,x 0) K t) , where T ( t− t 0,x 0 ) is an appropriate “leveling operator.” For this reason, the results obtained for the canonical solution (b) hold with only minor modifications for all other positive solutions. Toward the end of the paper it is pointed out that the negative of the functional F enters into relations analogous to those which restrict the stress functional in the thermodynamics of materials with fading memory; here K plays the role of the “strain,” and an appropriate “free-energy functional” is constructed. It is shown that, on solutions x of (a), the value ϕ( t) of the free-energy functional obeys the relations ϕ(t)= 1 2 x(t) 2−K(t)x(t) and ϕ ̇ (t) ⩽ − x(t) K ̇ (t) .