Modelling of a seasonally perturbed competitive three species impulsive system

Abstract

<abstract><p>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.</p></abstract>

A Seasonally Competitive M-Prey and N-Predator Impulsive System Modeled by General Functional Response for Integrated Pest Management

“…In practice, the timing for spraying the pesticides and when natural enemies should be release, as well as the quantity and frequency of the pesticides spray and the population of natural enemies to be released, are necessary to know in advance. Due to the discontinuous nature of the pesticides spray and the release of natural enemy events, the predator-prey system such as this kind can be modeled by using impulsive differential equations which have been widely adopted by mathematical ecologists [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. Since the pesticide is harmful to the pest, as well as to the natural enemies, they are ideal to deploy in two different time instances rather than for both to be applied at the same time.…”

“…Since the pesticide is harmful to the pest, as well as to the natural enemies, they are ideal to deploy in two different time instances rather than for both to be applied at the same time. Furthermore, much of the previous works consider two species, namely one prey and one predator, for modeling the dynamics of the predator-prey ecosystem, which is far from realistic [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. Although recent work has reported the study of three species system which considers two-prey and one-predator, as well as the one-prey and two-predator systems [2], the consideration of the effects due to the intra-and inter-species competitions in these studies are lacking [1][2][3][4][5][7][8][9][10][11][12][13][14][15][16][17][18][19][20].…”

“…Furthermore, much of the previous works consider two species, namely one prey and one predator, for modeling the dynamics of the predator-prey ecosystem, which is far from realistic [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. Although recent work has reported the study of three species system which considers two-prey and one-predator, as well as the one-prey and two-predator systems [2], the consideration of the effects due to the intra-and inter-species competitions in these studies are lacking [1][2][3][4][5][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. The co-existence of three or more species competing with each other often occurs in the real ecosystem, even when they are active in close range of geographical surroundings.…”

“…The survival and breeding of biological species are affected by the environmental change, which also influences their relationships such as their competitions, mutual co-operations, and predator-prey habitats [4,10]. Due to the diverse configurations in the biological systems, biological species' evolution in the natural environment has routinely been considered as an example of dynamic ecosystems, which can be investigated mathematically [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. For example, there is enough food in the warm seasons, such as in the spring and summer, that favors the animals and insects; thus, they enjoy high intrinsic growth rates.…”

“…For example, there is enough food in the warm seasons, such as in the spring and summer, that favors the animals and insects; thus, they enjoy high intrinsic growth rates. One objective of the present work is to extend previous research [1][2][3][5][6][7][8][9][10][11][12][14][15][16][17][18][19][20] by including environmental factors which modulate the prey's intrinsic growth rate periodically.…”