• Journal of the Korean Society for Industrial and Applied Mathematics
• /
• v.12 no.3
• /
• pp.139-151
• /
• 2008

We investigate a three species food chain system with Lotka-Volterra type functional response and impulsive perturbations. We find a condition for the local stability of prey or predator free periodic solutions by applying the Floquet theory and the comparison theorems and show the boundedness of this system. Furthermore, we illustrate some examples.

• Honam Mathematical Journal
• /
• v.30 no.3
• /
• pp.521-533
• /
• 2008

We investigate a periodically forced Lotka-Volterra type predator-prey system with impulsive perturbations - seasonal effects on the prey, periodic releasing of natural enemies(predator) and spraying pesticide at the same fixed times. We show that the solutions of the system are bounded using the comparison theorems and find conditions for the stability of a stable prey-free solution and for the permanence of the system.

• Kyungpook Mathematical Journal
• /
• v.48 no.3
• /
• pp.503-514
• /
• 2008

We investigate a three-species food chain system with Lotka-Volterra functional response and impulsive perturbations. In [23], Zhang and Chen have studied the system. They have given conditions for extinction of lowest-level prey and top predator and considered the local stability of lower-level prey and top predator eradication periodic solution. However, they did not give a condition for permanence, which is one of important facts in population dynamics. In this paper, we establish the condition for permanence of the three-species food chain system with impulsive perturbations. In addition, we give some numerical examples.

• The Pure and Applied Mathematics
• /
• v.17 no.3
• /
• pp.211-229
• /
• 2010

Various types of predator-prey systems are studied in terms of the stabilities of their steady-states. Necessary conditions for the existences of non-negative constant steady-states for those systems are obtained. The linearized stabilities of the non-negative constant steady-states for the predator-prey system with monotone response functions are analyzed. The predator-prey system with non-monotone response functions are also investigated for the linearized stabilities of the positive constant steady-states.

• Journal for History of Mathematics
• /
• v.26 no.2_3
• /
• pp.197-210
• /
• 2013

The late 18C Malthus studied population growth for the first time, Verhulst the logistic model in 19C and, after that, the study of the predation competition between two species resulted in the appearance of Lotka-Volterra model and modified model supported by Gause's experiment with bacteria. Instable coexistence equilibrium being found, Solomon and Holling proposed functional and numerical response considering limited abilities of predator on prey, which applied to Lotka Volterra model. Nicholson and Baily, considering the predation between host and parasitoid in discrete time, made a model. In 20C there were developed various models of disease dynamics with the help of mathematics and real data and named SIS, SIR or SEIR on the basis of dynamical phenomena.