Intra-Specific Density Dependent Effect of a Host-Parasitoid Interaction Model

Abstract

In this paper, the effect of intra-specific competition of host in a host-parasitoid model is investigated qualitatively by computer simulation. Many forms of complex dynamic are observed, including quasi-periodic, Hopf bifurcation reversal, and attractors crises. Furthermore, we obtain that the density- dependent effect may be a stabilizing factor. When hosts intra- specific competition stronger, the parameter region for persistent and stable interaction increases. I. INTRODUCTION The size of a natural population varies constantly and the variations may be small or large; sometimes regular, but in most cases irregular (1)-(4), so the dynamics of natural populations are very complicated (5), (6). The research area dealing with the complexities in the population dynamic models is the central issue in population ecology. In particular, the pioneering work in this field was initiated by May (7), (8). Now, the theory of single-population dynamics is quite well understood compared with the dynamics of interacting populations. Scientists have focused on studying interspecific interaction of natural populations whose generations are non- overlapping that can be modeled by difference equations. Difference equations describe how the population evolves in discrete time-step and can produce a much richer set of dynamic patterns than those observed in continuous-time models (9). Scientists have established many mathematical models to explain the dynamic behaviors of these interactions. It is not easy to analyze its global stability by qualitative method since the intrinsic nonlinearity, so people often study the dynamic complexity of host-parasitoid model by computer simulation. For different parameters and initial conditions, we can iterate the difference equations for thousands time steps and analyze the time series of population size to elucidate the regularity and mechanisms that hidden behind the population dynamics. Recently, many authors have adopted computer simulation to investigate the complexities of discrete-time host-parasitoid models. Kaitala and Heino (10) reported the dynamic com- plexity of host-parasitoid interaction with immunized and non- immunized host. Kaitala Ylikarjula and Heino (11), Tang and Chen (12) showed that many forms of complex dynam- ics were observed in host-parasitoid interaction model with Holling-type functional response. Xu and Boyce (13) also demonstrated the dynamic complexity of a mutual interference host-parasitoid model. All these research relied on a Logistic growth function to analyze the dyanmics of the host-parasitoid interaction and obtained some intriguing results. However, all these research are not to account for the discrepancy between dynamics predicted from these mathematical models is usually very intrigued and the dynamic behavior of real data is much simpler. Since this obvious discrepancy, many ecologists refused to accept the predictions from the deterministic mathematical models and prefer to accept environmental noise as the major driving force. In paper (14), the conclusion is that dynamic complexities are alleviated by Allee effect that is a kind of intra-specific interaction and have strengthened the utility of mathematical models in exploring populations. In this Paper, based on research of paper (15), the population dynamics effects by density dependent such as intra-specific competition will be extensively analyzed. In the next section, we will first propose the host-parasitoid model and give the explanation of all the parameters and variables. Then, the population dynamics will be fully investigated through numerical simulations. At last, we will conclude the results and give a short discussion.