Stability and Hopf bifurcation for a three-species food chain model with time delay and spatial diffusion

Top-predator interference and gestation delay as determinants of the dynamics of a realistic model food chain

“…Fig. 7(B) shows the Lyapunov exponent of Y with respect to the delay parameter j in a smaller range [12,4] for fixed s ¼ 4 . Positive Lyapunov exponent ðk) indicates that the system exhibits chaotic behavior [25,26].…”

“…Gopalsamy [3] and Kuang [1] studied the stability and oscillations in delay differential equations and its application to population dynamics. Ma et al [4] studied the stability and Hopf-bifurcation for a delayed three species Lotka-Volterra food chain model with time delay and spatial diffusion.…”

Analysis of an Impulsive One-Predator and Two-Prey System with Stage-Structure and Generalized Functional Response

“…Solution of (3), denoted by ( ) = ( 1 ( ), 2 ( ), 1 ( ), 2 ( )) , is piecewise and continuous function: : + → 4 + , ( ) is continuous on ( , ( + 1) ], ∈ + , and ( + ) = lim → + ( ) exists. Obviously, the global existence and uniqueness of solution of (3) are guaranteed by the smoothness of = ( 1 , 2 , 3 , 4 ), where denotes the mapping defined by right side of system (2). For more details refer to [20,21].…”

“…If examining the cases that there are two preys for a predator, then the above system cannot reflect the real behaviors of individuals accurately, so scholars proposed three-species predator-prey system. The relationship between species in three-species system may take many forms, such as one prey and two predators [1], a food chain [2,3], or two preys and one predator [4,5]. On the other hand, for predator-prey model, in description of the relationship between predator and prey, a crucial element is the classic definition of a predator's functional response.…”

“….). By use of impulsive differential equation theory and some analysis techniques, we aim to investigate the existence and global attractivity of predator-extinction periodic solution and the permanence of (2). Further, by numerical analysis, we try to find out the effects of impulsive and stage-structure on this system.…”