References Bhattacharyya, R. and Mukhopadhyay, B. (2006). Spatial dynamics of nonlinear prey-predator models with prey migration and predator switching, Ecological Complexity 3 (2): 160-169. Faria, T. (2001). Stability and bifurcation for a delayed predatorprey model and the effect of diffusion, Journal of Mathematical Analysis and Applications 254 (2): 433-463. Gao, S. J., Chen, L. S. and Teng, Z. D. (2008). Hopf bifurcation and global stability for a delayed predator-prey system with stage structure for predator, Applied Mathematics and Computation 202

of Vector Fields , Springer-Verlag, New York, NY. Hainzl, J. (1988). Stability and Hopf bifurcation in a predator-prey system with several parameters, SIAM Journal on Applied Mathematics 48 (1): 170-190. Harrison, G.W. (1986). Multiple stable equilibria in a predator-prey system, Bulletin of Mathematical Biology 48 (2): 137-148. He, Z. and Lai, X. (2011). Bifurcation and chaotic behavior of a discrete-time predator-prey system, Nonlinear Analysis: Real World Applications 12 (1): 403-417. Holling, C.S. (1965). The functional response of predators to prey

## Abstract

For the model of periodic chronic myelogenous leukemia considered by Pujo-Menjouet, Mackey et al., model consisting of two delay differential equations, the equation for the density of so-called “resting cells” was studied from numerical and qualitative point of view in several works. In this paper we focus on the equation for the density of proliferating cells and study it from a qualitative point of view.

certain threshold. The results were mathematically interesting, leading to both supercritical and subcritical Hopf bifurcations in the system, with a rich variety of bifurcations in the system. Here both the model and analysis are much simpler and yet the model is able to grasp all the difficulties presented in the first approach by Ajiraldi and Venturino [ 1 ]. The paper is organized as follows. In Section 2 we present the proposed mathematical model and in Section 3 classical techniques are used for its analysis, such as linear stability analysis, the Grob

## Abstract

In this paper, we investigate the combined effect of internal heating and time periodic gravity modulation in a viscoelastic fluid saturated porous medium by reducing the problem into a complex non-autonomous Ginzgburg-Landau equation. Weak nonlinear stability analysis has been performed by using power series expansion in terms of the amplitude of gravity modulation, which is assumed to be small. The Nusselt number is obtained in terms of the amplitude for oscillatory mode of convection. The influence of viscoelastic parameters on heat transfer has been discussed. Gravity modulation is found to have a destabilizing effect at low frequencies and a stabilizing effect at high frequencies. Finally, it is found that overstability advances the onset of convection, more with internal heating. The conditions for which the complex Ginzgburg-Landau equation undergoes Hopf bifurcation and the amplitude equation undergoes supercritical pitchfork bifurcation are studied.

References [1] SCHINASI, G. J.: Fluctuations in a dynamic, intermediate-run IS-LM model: applications of the Poincar´e-Bendixon theorem , J. Econom. Theory 28 (1982), 369-375. [2] KALDOR, N.: A model of the trade cycle , Economic J. 50 (1940), 69-86. [3] LIU, W. M.: Criterion of Hopf bifurcations without using eigenvalues , J. Math. Anal. Appl. 182 (1994), 250-256.

Biffurcations . The Publishing House of The Saint Petersburg University, Saint Petersburg, 1991. (In Russian) [5] GANDOLFO, G.: Economic Dynamics . Springer-Verlag, Berlin, 1997. [6] LIU, W. M.: Criterion of Hopf bifurcations without using eigenvalues , J. Math. Anal. Appl. 182 (1994), 250-256.

bifurcation and stability of periodic solutions for van der Pol equation with time delay, Nonlinear Analysis 62: 141-165. Źebrowski, J., Kuklik, P. and Baranowski, R. (2008). Relaxation oscillations in the atrium-a model, Proceedings of the 5th Conference of the European Study Group on Cardiovascular Oscillations, Parma, Italy, pp. 04:16-04:19. Zhou, X., Jiang, M. and Cai, X. (2011). Hopf bifurcation analysis for the van der Pol equation with discrete and distributed delays, Discrete Dynamics in Nature and Society, 2011: 1-8, Article ID: 569141.

Bibliography [1] H. N. Agiza, E. M. ELabbasy, H. EL-Metwally and A. A. Elsadany, Chaotic dynamics of a discrete prey-predator model with Holling type II, Nonlinear Analysis: Real World Applications 10 (2009), 116-129. [2] L. Edelstein-Keshet, Mathematical Models in Biology , Random House, New York, 1988. [3] S. Elaydi, Discrete Chaos: With Applications in Science and Engineering , Second Edition, Chapman & Hall/CRC, 2008. [4] J. Hainzl, Stability and Hopf bifurcation in a predator-prey system with several parameters, SIAM J. Appl. Math. 48 (1988), 170

fraction E * / N * of infected aphids decreases with a , since β γ E ∗ / N ∗ = β γ − q c − q ( γ + p ) / N ∗ . $$\begin{array}{} \displaystyle \beta\gamma E^*/N^* = \beta\gamma - qc - q(\gamma + p)/N^*. \end{array}$$ (11) The bifurcation diagram with the bifurcations we have seen so far is sketched in Figure 4 . There is no further bifurcation simply involving steady states, but Hopf bifurcations have not yet been ruled out. Fig. 4 Sketch of the transcritical bifurcations undergone by the system’s equilibria. 6 Steady states: stability The Jacobian matrix J for