A simple prey-predator-type model for the growth of tumor with discrete time delay in the immune system is considered. It is assumed that the resting and hunting cells make the immune system. The present model modifies the model of El-Gohary (2008) in that it allows delay effects in the growth process of the hunting cells. Qualitative and numerical analyses for the stability of equilibriums of the model are presented. Length of the time delay that preserves stability is given. It is found that small delays guarantee stability at the equilibrium level (stable focus) but the delays greater than a critical value may produce periodic solutions through Hopf bifurcation and larger delays may even lead to chaotic attractors. Implications of these results are discussed.

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This paper presents the use of real-valued Genetic Algorithm (GA) to evolve set of unknown parameters and initial conditions for tumor growth model using data extracted from El-Gohary [1]. The main focus of this work is to reach beyond the possibilities of traditional optimization methods in obtaining the far situated global optimum solutions with the help of arithmetic crossover and uniform mutation operators. Experimental results show the effectiveness of our approach by comparing the results obtained against the one mentioned in El-Gohary [1].

In this paper, a two species host-parasitoid model system is considered. The global dynamic behavior of the model is investigated through (local) stability results for its equilibriums and large time computer simulations. Many forms of complex dynamics such as chaos, periodic windows etc. are observed. The Hopf point and attractor crises exist for different set of parameter values.