Seasonal Effects on a Beddington-DeAngelis Type Predator-Prey System with Impulsive Perturbations

and Applied Analysis 3 following predator-prey system with adding periodic constant impulsive immigration of the predator regarded as natural enemy of the prey pest to system 1.3 and spraying pesticides harvesting on all species at the same times x′ t rx t ( 1 − x t K ) − ax t y t by t x t c λx t sin ωt , y′ t −dy t eax t y t by t x t c , t / nτ, x t ( 1 − p1 ) x t , t nτ, y t ( 1 − p2 ) y t q, ( x 0 , y 0 ) ( x0, y0 ) , 1.4 where τ is the period of the impulsive immigration or stock of the predator, 0 ≤ p1, p2 < 1 present the fraction of the prey and the predator which die due to the harvesting or pesticides, and so forth, and q > 0 is the size of immigration or stock of the predator. If we take b 0, system 1.4 can be expressed as the Holling-type II predator-prey system with impulsive perturbations and seasonal effects as follows: x′ t rx t ( 1 − x t K ) − ax t y t x t c λx t sin ωt , t / nτ, y′ t −dy t eax t y t x t c , x t ( 1 − p1 ) x t , t nτ, y t ( 1 − p2 ) y t q, ( x 0 , y 0 ) ( x0, y0 ) . 1.5 While if c 0, then system 1.4 can be expressed as the ratio-dependent predator-prey system with impulsive perturbations and seasonal effects as follows: x′ t rx t ( 1 − x t K ) − ax t y t by t x t λx t sin ωt , t / nτ, y′ t −dy t eax t y t by t x t , x t ( 1 − p1 ) x t , t nτ, y t ( 1 − p2 ) y t q, ( x 0 , y 0 ) ( x0, y0 ) . 1.6 We will investigate system 1.4 together with systems 1.5 and 1.6 . Impulsive differential equations such as 1.4 are found in almost every domain of applied science and have 4 Abstract and Applied Analysis