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HOPF BIFURCATION ANALYSIS OF A PREDATOR-PREY MODEL WITH TIME-DELAY


An n. of D i?. Eqs . 28:1(2012) , 15-20 HOPF BIFURCATION ANALYSIS OF A PREDATOR-PREY MODEL WITH TIME-DELAY Hongbing C he n, Wansheng He, Xiaoke Sun ?? (Sch oo l of Ma th. a nd Sta tistics, Tia nsh ui No rm al U niversity, T iansh ui 7 41 0 01 , Ga nsu, E-m a il: chenh ong bing5 1 8@1 26 .com ( H . C hen) ) A bs tra ct In this pap er, a predat o r-prey m o del o f th ree sp ecies is inve st ig a ted. B y int ro ducing a dela y a s a bifurca tion pa ra m ete r, it is fo und th at H o pf bifurca tion o ccurs w hen τ cro sses so m e crit ica l values. Key wo rds H o pf bifu rca tio n; st ability; t im e delay; preda to r-prey syst em ; e qu ilibriu m p oint 20 00 M athem atic s Sub ject C lassi?c ation 3 4 D12 1 Introduction Th e dy nam ic al re lation ship b e twee n p red ators an d p rey h as lon g b een and will c ontinue to be one of d ominant th eme s in both ecology an d math ematical ec ology du e to its un iversal ex iste nc e an d im p ortan ce . In th is pap er, we c onside r a p red ator- pre y mo de l with time d elay ? ˙ 1 ( t ) = x1 ( t)[ r 1 ? a1 x 1 ( t ? τ ) + b 1 x 2 ( t ? τ ) ? c 1 x 3 ( t ? τ )] , ? ? x x ˙ 2 ( t ) = x2 ( t)[ r 2 ? a2 x 2 ( t ? τ ) + b 2 x 1 ( t ? τ ) ? c 2 x 3 ( t ? τ )] , (1 . 1) ? ? x ˙ 3 ( t ) = x3 ( t)[ ? r3 + b3 x1 ( t ? τ ) + b4 x2 ( t ? τ ) ? a3 x 3 ( t ? τ )] , wh ere x 1 ( t) an d x 2 ( t ) de note the p opu lation de nsity of p re y, an d x3 ( t) is the p opu lation de nsity of pre dator. Th e parame ters r 1 > 0 an d r 2 >

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