Abstract
In this paper, we present new results on deterministic sudden changes and stochastic fluctuations’ effects on the dynamics of a two-predator one-prey model. We purpose to study the dynamics of the model with some impacting factors as the problem statement. The methodology depends on investigating the seasonality and stochastic terms which make the predator-prey interactions more realistic. A theoretical analysis is introduced for studying the effects of sudden deterministic changes, using three different cases of sudden changes. We show that the system in a good situation presents persistence dynamics only as a stable dynamical behavior. However, the system in a bad situation leads to three main outcomes as follows: first, constancy at the initial conditions of the prey and predators; second, extinction of the whole system; third, extinction of both predators, resulting in the growth of the prey population until it reaches a peak carrying capacity. We perform numerical simulations to study effects of stochastic fluctuations, which show that noise strength leads to an increase in the oscillations in the dynamical behavior and became more complex and finally leads to extinction when the strength of the noise is high. The random noises transfer the dynamical behavior from the equilibrium case to the oscillation case, which describes some unstable environments.
Highlights
IntroductionSeasonality is an important factor, which plays a vital role in describing the changes and fluctuations in ecological systems with predator-prey interactions [12,13,14,15,16,17,18]
We aim to investigate a cosinusoidal function in a Holling type I two-predator one-prey model, in order to study how sudden changes of the dynamics will effect on the dynamical behavior of the model
We investigated the seasonality effects in a Holling type I two-predator one-prey model, which can more realistically describe the species of interaction more realistic. e nonautonomous models are transferred to autonomous models by approximating the model to particular cases representing sudden changes, so the situations are classified to bad and good situations, according to the surrounding circumstances
Summary
Seasonality is an important factor, which plays a vital role in describing the changes and fluctuations in ecological systems with predator-prey interactions [12,13,14,15,16,17,18]. Deterministic models are useful, due to their ability to follow them through mathematical analysis, and they are an important mechanism for describing stable environments. Stochastic predator-prey models and their dynamics have been studied by some researchers [20,21,22,23,24,25]. We aim to investigate a cosinusoidal function in a Holling type I two-predator one-prey model, in order to study how sudden changes of the dynamics will effect on the dynamical behavior of the model. E methodology of arrays: Array 1: adding the stochastic term Array 2: adding the seasonality function Array 3: using the approximation method Array 4: theoretical analysis Array 5: numerical simulation
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