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A predator-prey system involving group defense: A connection matrix approach. (English) Zbl 0724.34015
In this paper the authors give a concrete demonstration of how the connection matrix techniques can be used to analyse a one-parameter family of differential equations, which arise from a predator-prey model in which the prey exhibits group defense. They show how to compute connection matrices and how to apply those matrices to classify the structure of solutions. They prove the existence of local and global bifurcations and they demonstrate how to ignore certain bifurcations. They do not claim any new results, rather it is their techniques which are novel.
Reviewer: M.Lizana (Caracas)

34C23 Bifurcation theory for ordinary differential equations
92D25 Population dynamics (general)
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
Full Text: DOI
[1] Conley, C.C., Isolated invariant sets and the Morse index, () · Zbl 0397.34056
[2] Franzosa, R., The connection matrix theory for Morse decompositions, Trans. am. math. soc., 311, 561-592, (1989) · Zbl 0689.58030
[3] Freedman, H.I.; Wolkowicz, G.S.K., Predator-prey systems with group defense: the paradox of enrichment revisited, Bull. math. biol., 48, 493-508, (1986) · Zbl 0612.92017
[4] \scHattori H. & \scMischaikow K., On the existence of intermediate magnetohydrodynamic shock waves (to appear). · Zbl 0695.76030
[5] Hsu, S.B., On global stability of a predator-prey system, Math. biosci., 39, 1-10, (1978) · Zbl 0383.92014
[6] McCord, C., The connection map for attractor-repeller pairs, Trans. am. math. soc., 307, 195-203, (1988) · Zbl 0646.34056
[7] Mischaikow, K., Existence of generalized homoclinic orbits for one-parameter families of flows, Proc. am. math. soc., 103, 59-68, (1989) · Zbl 0661.34038
[8] Mischaikow, K., Transition matrices, Proc. R. soc. edinb., 112A, 155-175, (1989) · Zbl 0677.34046
[9] Mischaikow, K.; Wolkowicz, G.S.K., A connection matrix approach illustrated by means of a predator-prey model involving group defense, (), 676-710
[10] Reineck, J., Connecting orbits in one-parameter families of flows, Engng theor. dyn. syst., 8*, 359-374, (1988) · Zbl 0675.58034
[11] Salamon, D., Connected simple systems and the Conley index of isolated invariant sets, Trans. am. math. soc., 291, 1-41, (1985) · Zbl 0573.58020
[12] Smoller, J., Shock waves and reaction diffusion equations, (1983), Springer New York · Zbl 0508.35002
[13] Wolkowicz, G.S.K., Bifurcation analysis of a predator-prey system involving group defense, SIAM J. appl. math., 48, 1-15, (1988) · Zbl 0657.92015
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