<abstract xmlns="http://www.w3.org/1999/xhtml">We present a qualitative population growth analysis approach using the Pearl logistic population growth differential equation for a population with the intensity of birth <italic>a</italic>, stationary size <italic>K</italic> to instances of fast small random population size extractions proportional to εξ<italic>n</italic> at random time moments τ<italic>n</italic>, where ε is the small positive parameter. Assuming that the intervals τ<italic>n</italic> − τ<italic>n</italic>−1 are independent identically exponentially distributed random variables with the parameter λ / ε, and that ξ<italic>n</italic> are the independent identically distributed positive random variables with the mean <italic>h</italic> and the variance <italic>b</italic>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{<italic>x</italic>(<italic>t</italic>)} of the population growth and the stochastic Ito differential equation for the normalised deviations (𝔼{<italic>x</italic>(<italic>t</italic>)} − <italic>x</italic>(<italic>t</italic>))ε−1/2. Assuming that the difference <italic>a</italic> − λ<italic>h</italic> = ε<italic>c</italic> is sufficiently small we derive the stochastic differential equation for the scaled population growth in accelerated time ε−1<italic>Kx</italic>(<italic>t</italic> / ε)and prove that under condition 2<italic>c</italic> &lt; λ(<italic>h</italic>2+<italic>b</italic>2) the population disappears with probability one, otherwise the distribution of the scaled population size with increasing time tends to the Gamma-distribution Γ(<italic>k</italic>,<italic>q</italic>) with the shape <italic>k</italic> = 2<italic>c</italic>/λ(<italic>h</italic>2 + <italic>b</italic>2) and the scale θ = λ(<italic>h</italic>2 + <italic>b</italic>2)/2<italic>c</italic>.</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 εξn at random time moments τn, where ε is the small positive parameter. Assuming that the intervals τn − τn−1 are independent identically exponentially distributed random variables with the parameter λ / ε, and that ξn are the independent identically distributed positive random variables with the mean h and the variance b2, 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−1Kx(t / ε)and prove that under condition 2c < λ(h2+b2) 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(h2 + b2) and the scale θ = λ(h2 + b2)/2c.

Dynamics of a Population Subject to Impulse Type Random Loss

Department of Theory of Probability and Mathematical Statistics

Proceedings of the Latvian Academy of Sciences. Section B. Natural, Exact, and Applied Sciences.

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"Dynamics of a Population Subject to Impulse Type Random Loss"}

We present a qualitative population growth analysis approach using the Pearl logistic population growth differential equation for a population with the...

Dynamics of a Population Subject to Impulse Type Random Loss

Published Online: Sep 09, 2017

Page range: 298 - 302

Received: Nov 10, 2016

Accepted: Apr 02, 2017

DOI: https://doi.org/10.1515/prolas-2017-0050

Keywords
population dynamics, stochastic equations, diffusion approximation

© 2017 Jevgeņijs Carkovs et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Dynamics of a Population Subject to Impulse Type Random Loss

Published Online: Sep 09, 2017

Page range: 298 - 302

Received: Nov 10, 2016

Accepted: Apr 02, 2017

DOI: https://doi.org/10.1515/prolas-2017-0050

Keywordspopulation dynamics, stochastic equations, diffusion approximation

© 2017 Jevgeņijs Carkovs et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
population dynamics, stochastic equations, diffusion approximation