Dynamics of a Population Subject to Impulse Type Random Loss

Abstract

The logistic growth model is the most popular one in population dynamics, and considers the birth and death rates typical to the species as well as the effects of different mechanisms of consumption of resources on the dynamics of the biological system. For qualitative analysis of the behaviour of an ecosystem the ordinary differential equation (Otto and Day, 1995):is commonly used, where the parameter a describes growth rate, i.e. fertility, and the parameter h determines the intensity of extraction of specimens from ecosystem (as an example, the intensity of fishing in the case of fish populations in fisheries, which we use here, and further as an illustrative example scenario). Parameter K corresponds to stationary population size in the case of an absence of fishing, i.e., stationary solution of the equation (1) if h = 0. As K -1 describes the rate of reduction of the population size, this coefficient may be considered as the death rate of the population. Let us note that the stationary solution of the equation (1)differs from K, and solutions of (1) tend exponentially to x st only if a > h. If this inequality is not true, the solution of (1) tends exponentially to zero. This means that the population dies out under such a fishing regime. If the fishing rate h is equal to the birth rate a, solutions of (1) also tend to zero, albeit not with an exponential velocity. This means that in order for fishing to be sustainable, it is necessary for the birth rate a to be greater than the fishing rate h. It should be noted that it is not just the size of the population that determines the fishing regime. Using the finite difference method, we see that the amount of fishing at time t should be approximately equal to hx(t)Dt. However, it may be the case that the entire volume of the population x(t) may not be suitable for fishing at the time t, and we can only rely on the average value of fishing hx(t)Dt. Sometimes it is possible to use a different fishing strategy: wait for a time period d, which, generally speaking, is a random time dependent variable, and only then remove the necessary amount of the population. Therefore, deterministic fishery strategies do not allow us to predict the dynamics of the population size and/or to determine fishing mode, especially if a » h. In this case, construction and exploration of mathematical models PROCEEDINGS OF THE LATVIAN ACADEMY OF SCIENCES. Section B, Vol. 71 (2017) ) the population disappears with probability one, otherwise the distribution of the scaled population size with increasing time tends to the Gamma-distribution G(k,q) with the shape k = 2c/l(h 2 + b 2 ) and the scale q = l(h 2 + b 2 )/2c.