Dynamics of a Population Subject to Impulse Type Random Loss

Abstract

Abstract We present a qualitative population growth analysis approach using the Pearl logistic population growth differential equation for a population with the intensity of birth a, stationary size K to instances of fast small random population size extractions proportional to εξ n at random time moments τ n , where ε is the small positive parameter. Assuming that the intervals τ n − τ n−1 are independent identically exponentially distributed random variables with the parameter λ / ε, and that ξ n are the independent identically distributed positive random variables with the mean h and the variance b 2, we analyse the population dynamics and the population asymptotic behaviour. We propose a probabilistic limit theorem based stochastic approximation algorithm for the qualitative analysis of the above model on any finite time interval. At first we derive the linear differential equation for mathematical expectation 𝔼{x(t)} of the population growth and the stochastic Ito differential equation for the normalised deviations (𝔼{x(t)} − x(t))ε−1/2. Assuming that the difference a − λh = εc is sufficiently small we derive the stochastic differential equation for the scaled population growth in accelerated time ε−1 Kx(t / ε)and prove that under condition 2c < λ(h 2+b 2) the population disappears with probability one, otherwise the distribution of the scaled population size with increasing time tends to the Gamma-distribution Γ(k,q) with the shape k = 2c/λ(h 2 + b 2) and the scale θ = λ(h 2 + b 2)/2c.