Abstract
The population of biological species in the ecosystem is known sensitive to the periodic fluctuations of seasonal change, food resources and climatic conditions. Research in the ecological management discipline conventionally models the behavior of such dynamic systems through specific impulsive response functions, but the results of such research are applicable only when the environments conform exactly to the conditions as defined by the specific response functions that have been implemented for specific scenarios. This means that the application of previous work may be somewhat limited. Moreover, the intra and inter competitions among species have been seldom studied for modelling the prey-predator ecosystem. To fill in the gaps this paper models the delicate balance of two-prey and one-predator system by addressing three main areas of: ⅰ) instead of using the specific impulse response this work models the ecosystem through a more general response function; ⅱ) to include the effects due to the competition between species and ⅲ) the system is subjected to the influences of seasonal factors. The seasonal factor has been implemented here in terms of periodic functions to represent the growth rates of predators. The sufficient condition for the local and global asymptotic stability of the prey-free periodic solution and the permanence of the system have been subsequently obtained by using the Comparison techniques and the Floquet theorems. Finally, the correctness of developed theories is verified by numerical simulation, and the corresponding biological explanation is given.
Highlights
The modelling of the dynamics in the predator-prey ecosystem research has been by large concerning with two species system, without much consideration on the effects due to the intra and inter species competitions [1,2,3,4]
To model the rich dynamics of situations like this will need to include the competitions between species, so to understand the balance of the biological community better even for small local geographical areas of ecosystem
The present work reveals that the dynamic system of the model is more complex than the one without taking the intra- and inter-species competition into consideration
Summary
The modelling of the dynamics in the predator-prey ecosystem research has been by large concerning with two species system, without much consideration on the effects due to the intra and inter species competitions [1,2,3,4]. Motivated by previous work in [1,29] and [19], this paper attempts to establish the modelling of the three species ecosystem by deploying impulsive control strategies at different instances of time period through a generalize functional response of gi(u1(t), u2(t), v(t)), and to compound the effects due to the periodic variation of the prey’s intrinsic growth rate, here we mainly talk about the two-prey and one-predator ecosystem: t du1(t) dt. We obtain the ecosystem (1.1)’s positive periodic solution about prey-free (0, 0, v (t))
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