Details

Autor(en) / Beteiligte

• Öztürk, I.
• Bozkurt, F.
• Gurcan, F.

Titel

Stability analysis of a mathematical model in a microcosm with piecewise constant arguments

Ist Teil von

• Mathematical biosciences, 2012-12, Vol.240 (2), p.85-91

Ort / Verlag

United States: Elsevier Inc

Erscheinungsjahr

2012

Quelle

MEDLINE

Beschreibungen/Notizen

• ► A bacteria population in a microcosm was modeled. ► Local and global stability analysis of the model basin under specific conditions was investigated. ► For a bacteria population at low density, an Allee function at time t was incorporated to the model. ► To show the consistence of a bacteria population with the constructed models, the informations in [8] was used. ► The example and the discussion section have shown the discrepancy of both models. In this paper, we have modeled a population density of a bacteria species in a microcosm by using a differential equation, (A)dx(t)dt=rx(t){1-αx(t)-β0x(〚t〛)-β1x(〚t-1〛)},where t⩾0, the parameters r, α, β0 and β1 denote positive numbers and 〚t〛 denotes the integer part of t∈[0, ∞). First, to obtain the local and global behaviors, the boundedness character and the periodic nature of the population density for bacteria, discrete solutions of differential Eq. (A) is investigated. Examinations of the stability characterization of (A) show that increasing of the population growth rate decreases the local stability of the positive equilibrium point. Due to this result we need to consider a second approximation to obtain stability of population density. This can be performed at low density by incorporating an Allee function to (A) at time t. For the theoretical results obtained here we give an example by taking some parameter values from experimental data of bacteria populations [8] and show that the experimental and theoretical results for both models with and without Allee effect are in good agreement.