A Logistic-Harvest Model with Allee Effect under Multiplicative Noise

Abstract

This work is devoted to the study of a stochastic logistic growth model with and without the Allee effect. Such a model describes the evolution of a population under environmental stochastic fluctuations and is in the form of a stochastic differential equation driven by multiplicative Gaussian noise. With the help of the associated Fokker-Planck equation, we analyze the population extinction probability and the probability of reaching a large population size before reaching a small one. We further study the impact of the harvest rate, noise intensity, and the Allee effect on population evolution. The analysis and numerical experiments show that if the noise intensity and harvest rate are small, the population grows exponentially, and upon reaching the carrying capacity, the population size fluctuates around it. In the stochastic logistic-harvest model without the Allee effect, when noise intensity becomes small (or goes to zero), the stationary probability density becomes more acute and its maximum point approaches one. However, for large noise intensity and harvest rate, the population size fluctuates wildly and does not grow exponentially to the carrying capacity. So as far as biological meanings are concerned, we must catch at small values of noise intensity and harvest rate. Finally, we discuss the biological implications of our results.

On the stability of a time‐varying single‐species harvesting model with Allee effect

“…Thereafter, many researchers have used oscillating carrying capacities to describe seasonal changes, for example, Leach and Andriopoulos [10] and Rogovchenko and Rogovchenko [11] and references therein. Lastly, special attention has to be paid to the harvesting strategies because they may sharply affect the long-term behavior of the population [12][13][14][15]. The relative-rate harvesting occurs when the number of population caught per time is proportional to the total population size, whereas the constant-rate harvesting occurs when the number of population is caught at a constant rate per unit time independently of the size of the population [16,17].…”

“…Among of the most influential dynamical behaviors of populations with logistic equations is the Allee effect which is a phenomenon characterized by reduced per capita population growth rate at low population sizes [15,[18][19][20]. This phenomenon is nonlinear, caused by various biological factors, and is associated with the increase of population extinction risk [19,21].…”

“…) (N (t) − L (t)), t ≥ t 0 , population density N(t), time-varying capacity K ∈ C 0 (R, R), Allee threshold L ∈ C 0 (R, R), harvest term 𝜇 ∈ C 0 (R, R), and a function r ∈ C 0 (R, R) that is related to the intrinsic growth rate. When r, K, L, and 𝜆 are constant functions, sufficient stability conditions for the prior systems have been mathematically formulated and rigorously proven in the literature [15,18,20]. However, the time-varying case has not been considered in the literature because of its complexity.…”

Stochastic Bifurcation in Single-Species Model Induced by α-Stable Levy Noise

“…where X is the population size at time t in a given area or place, s > 0 is the population growth rate when the population is small, and carrying capacity M > 0 is usually determined by the available sustaining resource. The carrying capacity is often assumed to be a constant [9,10]. In this paper we assume a state-dependent carrying capacity M(X)( cf.…”

“…Our previous papers [9,10] assumed a constant carrying capacity in Eq. (1.8) and investigated the extinction probability of the population.…”