Periodic Solutions of the N -Preys and M -Predators Model with Variable Rates on Time Scales
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DOI: 10.1007/s13226-020-0443-3
PERIODIC SOLUTIONS OF THE N -PREYS AND M -PREDATORS MODEL WITH VARIABLE RATES ON TIME SCALES Shekhar Singh Negi, Syed Abbas and Muslim Malik School of Basic Sciences, Indian Institute of Technology Mandi, Kamand (H.P.) 175 005, India e-mail: [email protected] (Received 28 April 2017; after final revision 24 January 2019; accepted 14 May 2019) In this paper, we establish the existence of periodic solution of a delayed predator-prey model with M -predators and N -preys over the time scales. We derive sufficient conditions for the existence of a periodic solution with the help of continuation theorem of coincidence degree theory. At the end, we give an example with numerical simulations to illustrate our analytical findings. Key words : Time scale; prey-predator model; continuation theorem; coincidence degree; periodic solution. 2010 Mathematics Subject Classification : 34N05, 93A30, 47H11.
1. I NTRODUCTION In the past few decades, mathematical ecology has seen extensive progress, especially in population dynamics. It is the study of how a population changes over time and space or the relationship between the population and its environment. In the study of population dynamics, the interaction between a pair of predators and prey influences the population growth of both the species, i.e., when two or more species interact to each other their growth totally depends on each other species. For the understanding of dynamic behavior of the species, several population models are considered by many authors. For example, in 1989, the asymptotic behavior of two-dimensional Lotka-Volterra model has been studied by Ma and Wendi [18], and in 1995, two-predators and one-prey periodic Lotka-Volterra system was investigated by Zhonghua and Lansun, see [28]. In 1999, A Lotka-Volterra model with competition among predator species and among prey species or M -predators and N -preys was simultaneously
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SHEKHAR SINGH NEGI, SYED ABBAS AND MUSLIM MALIK
considered by Yang and Rui [23], and then studied existence and uniqueness of the periodic solution of the system µ ¶ N M P P aik (t)xk (t) − cil (t)yl (t) , i = 1, 2, · · · , N, x˙ i (t) = xi (t) bi (t) − k=1 l=1 µ ¶ M N P P y˙ j (t) = yj (t) −rj (t) − ejl (t)yl (t) + djk (t)xk (t) , j = 1, 2, · · · , M, l=1
(1.1)
k=1
where xi (t) and yj (t) denoted the numbers of prey and predator species respectively and bi (t), rj (t), aik (t), cil (t), djk (t) and ejl (t) (i, k = 1, 2, · · · , N and l, j = 1, 2, · · · , M ) denoted the coefficients, which are non-negative continuous periodic functions defined on t ∈ (−∞, ∞), where the coefficient bi is the natural growth rate of prey species, rj is the death rate of the predator species, aik measures the amount of the emulation between the prey species, ejl measures the amount of the emulation ∆ d between the predator species, and the constant k¯ij = ij denotes conversing prey species into new cij
individual of predator species. Many times it is necessary to consider the delayed models because
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