Non-autonomous Logistic Equation Research Articles

Precise spectral asymptotics for nonautonomous logistic equations of population dynamics in a ball

We consider the semilinear elliptic eigenvalue problem −Δu + k(|x|)up = λu, u > 0 in BR, u = 0 on ∂BR, where p > 1 is a constant, BR : = {x ∈ RN : |x| < R}(N ≥ 1), and λ > 0 is a parameter. We investigate the global structure of the branch of (λ, uλ) of bifurcation diagram from a point of view of L2‐theory. To do this, we establish a precise asymptotic formula for λ = λ(α) as α → ∞, where .

A note on nonautonomous logistic equation with random perturbation

This paper discusses a randomized nonautonomous logistic equation d N ( t ) = N ( t ) [ ( a ( t ) − b ( t ) N ( t ) ) d t + α ( t ) d B ( t ) ] , where B ( t ) is 1-dimensional standard Brownian motion. We show that E [ 1 N ( t ) ] has a unique positive T-periodic solution E [ 1 N p ( t ) ] provided a ( t ) , b ( t ) , and α ( t ) are continuous T-periodic functions, a ( t ) > 0 , b ( t ) > 0 and ∫ 0 T [ a ( s ) − α 2 ( s ) ] d s > 0 .

Global attractivity of a nonautonomous logistic difference equation with delay

Consider the nonautonomous delay logistic difference equation Δy n= 1− y n−k n λ where { p n } n≥0 is a sequence of nonnegative real numbers, { k n } n≥0 is a sequence of positive integers, n − k n is nondecreasing, lim n →∞( n − k n ) = ∞, and λ is a positive constant. We obtain new sufficient conditions for the global attractivity of the equilibrium λ of equation (1.1), which improve some recent results established in [1].

Extinction in nonautonomous competitive Lotka-Volterra systems

It is well known that for the two species autonomous competitive Lotka-Volterra model with no fixed point in the open positive quadrant, one of the species is driven to extinction, whilst the other population stabilises at its own carrying capacity. In this paper we prove a generalisation of this result to nonautonomous systems of arbitrary finite dimension. That is, for thennspecies nonautonomous competitive Lotka-Volterra model, we exhibit simple algebraic criteria on the parameters which guarantee that all but one of the species is driven to extinction. The restriction of the system to the remaining axis is a nonautonomous logistic equation, which has a unique solutionu(t)u(t)that is strictly positive and bounded for all time; see Coleman (Math. Biosci.45(1979), 159–173) and Ahmad (Proc. Amer. Math. Soc.117(1993), 199–205). We prove in addition that all solutions of thenn-dimensional system with strictly positive initial conditions are asymptotic tou(t)u(t).

Size Structured Competition Model with Periodic Coefficients

A model of Diekmann et al. [14] for two size-structured populations competing for a single resource is extended in this paper to allow for periodic variation in parameter values (having a common period). The hyperbolic partial differential equation model is reduced to systems of ordinary differential equations describing the dynamic of two size-structured species competing for a single unstructured resource. The existence of nontrivial periodic solutions of those systems is considered. Under fairly general conditions, global bifurcation techniques as used by Gushing are used to show the existence of a continuum of solutions that bifurcate from a noncritical periodic solution of the reduced single species system. The positivity and the stability of the bifurcating branch solutions are studied. The stability is shown to depend on the stability of the solutions of the reduced system and the direction of bifurcation. The results and techniques can be generalized to the case of n, n ≥ 1, size-structured species competing for a single resource. It is also shown that the average size of individuals of each species is governed by a nonautonomous logistic equation whose solution under fairly general conditions approaches the unique periodic solution of the classical periodic logistic equation.

Periodic solutions to nonautonomous difference equations

A technique is presented for determining when periodic solutions to nonautonomous periodic difference equations exist. Under certain constraints, stable periodic solutions can be guaranteed to exist, and this is used to compare the analogous behavior of a nonautonomous periodic hyperbolic difference equation to that of the nonautonomous periodic Pearl-Verhulst logistic differential equation.

Non-autonomous logistic equations as models of populations in a deteriorating environment

Abstract The non-autonomous logistic equation d x(t) d t = r(t)x(t)[1 − x(t) K(t) ] is studied under conditions that include an environment which is completely deteriorating. In this setting, when the population's growth rate, r, is large on the average, solutions track the environment with a consequent extinction of the population. However, when both r and rK−1 are small in the sense that they are in L1[0,∞) then an asymptotic equivalence, where all solutions tend to positive limits as t approaches infinity, results and the population is persistent, independent of initial density. The asymptotic equivalence produces an unreasonable overshoot of carrying capacity which leads to concern about employing the logistic equation in the above form as a population model when growth rates are close to zero. A re-interpretation of the parameters of the logistic equation leads to the alternative logistic formulation d x(t) d t = x(t)[r(t) − c B(t) x(t)], (c > 0) . A biological interpretation of the parameters is presented and this equation is compared with the classical logistic model in the case where the parameters are constant. If the alternative logistic model is applied in a situation with time-varying parameters, then a deteriorating environment always leads to extinction of the population regardless of the behavior of r. Similarly, a growth rate which is small on the average results in extinction regardless of the behavior of B. Furthermore, r and B have limiting values as t approaches infinity then so does x and the terminal value of x is equal to the terminal value of the carrying capacity of the population. In general, the alternative formulation seems to be the more reasonable model in situations where perturbations lead to severe decreases in environmental quality and growth rates.

On the optimal choice of r for a population in a periodic environment

For each choice of the intrinsic growth rate r and the environmental carrying capacity K as positive, bounded, periodic functions with period p, the nonautonomous logistic equation, x ̇ (t)=r(t)x(t) 1− x(t) K(t) , possesses an asymptotically stable, positive, periodic solution x ∗ with period p. If K is not a constant function, but is piecewise continuous on [0, p], then the minimum and maximum values of x ∗ are related as follows to the extrema of K: K inf <x ∗ min <x ∗ max <K sup . The following question is treated here: For each specification of the function K, which functions r come close to maximizing x ∗ min? It is shown that if r is expressed in the form r( t)= δγ( t) with δ>0 and γ a positive p-periodic function whose average value is unity, then, for δ small enough, x ∗( t) is approximately equal to a number x̃(γ, K) which is independent of t; in fact, for each t, lim δ→0 x ∗(t)=p ∫ 0 p γ(τ) K(τ) dτ −1= x ̃ (γ,K) . By an appropriate choice of γ, the number x̃(γ, K) can be made arbitrarily close to K sup. The appropriate choices are those for which γ is approximately a “Dirac function” with its support concentrated at times for which K is close to K sup.

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

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) .