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Periodic solutions for a semi-ratio-dependent predator-prey system with Holling IV functional response. (English) Zbl 1201.34073
The authors consider nonautonomous semi-ratio-dependent predator-prey systems $$\dot x_1 = (r_1 (t)-a_{11}(t)x_1)x_1 - f(t,x_1)x_2, \quad \dot x_2= \left ( r_2 (t)-a_{21} (t)\frac{x_2}{x_1} \right ) x_2$$ with initial conditions $x_i (0) >0$, $i=1,2$ and where the nonmonotonic functional response $f(t,x_1)$ is given by $f(t,x_1) = \frac{a_{12}(t)x_1}{m^2 + nx_1 + x_1^2}$. Here, $x_1$ and $x_2$ denote the density of the prey and the predator, respectively, $m \neq 0$, $n \geq 0$ and $r_i (t)$, $a_{ij} (t)$, $i,j= 1,2,$ are continuous, positive and $\omega$-periodic functions. First, by using a continuation theorem of coincidence degree theory, the authors discuss the existence of positive $\omega$-periodic solutions of the previous system. Then, by constructing a Lyapunov function, they establish a sufficient condition for the uniqueness and global asymptotic stability of such positive periodic solution.

34C60Qualitative investigation and simulation of models (ODE)
34C25Periodic solutions of ODE
92D25Population dynamics (general)
47N20Applications of operator theory to differential and integral equations
34D20Stability of ODE
Full Text: DOI
[1] Andrews, J.F.: A mathematical model for the continuous culture of microorganisms utilizing inhabitory substrates. Biotechnol. Bioeng. 10, 707--723 (1986) · doi:10.1002/bit.260100602
[2] Bohner, M., Fan, M., Zhang, J.M.: Existence of periodic solutions in predator-prey and competition dynamic systems. Nonlinear Anal. 7, 1193--1204 (2006) · Zbl 1104.92057 · doi:10.1016/j.nonrwa.2005.11.002
[3] Bush, A.W., Cook, A.E.: The effect of time delay and growth rate inhibition in the bacterial treatment of wastewater. J. Theor. Biol. 63, 385--395 (1976) · doi:10.1016/0022-5193(76)90041-2
[4] Collings, J.B.: The effects of the functional response on the bifurcation behavior of a mite predator-prey interaction model. J. Math. Biol. 36, 149--168 (1997) · Zbl 0890.92021 · doi:10.1007/s002850050095
[5] Cushing, J.M.: Periodic time-dependent predator-prey system. SIAM J. Appl. Math. 32, 82--95 (1977) · Zbl 0348.34031 · doi:10.1137/0132006
[6] Fan, M., Wang, Q.: Periodic solutions of a class of nonautonomous discrete time semi-ratio-dependent predator-prey system. Discrete Continuous Dyn. Syst. B 4(3), 563--574 (2004) · Zbl 1100.92064 · doi:10.3934/dcdsb.2004.4.563
[7] Fan, Y.H., Li, W.T., Wang, L.L.: Periodic solutions of delayed ratio-dependent predator-prey model with monotonic and nonmonotonic functional response. Nonlinear Anal. 5(2), 247--263 (2004) · Zbl 1069.34098 · doi:10.1016/S1468-1218(03)00036-1
[8] Gaines, R.E., Mawhin, J.L.: Coincidence Degree and Nonlinear Differential Equations. Springer, Berlin (1977) · Zbl 0339.47031
[9] Gopalsamy, K.: Stability and Oscillations in Delay Differential Equations of Population Dynamics. Kluwer Academic, Dordrecht/Norwel (1992) · Zbl 0752.34039
[10] Hsu, S.B., Huang, T.W.: Global stability for a class of predator-prey systems. SIAM J. Appl. Math. 55, 763--783 (1995) · Zbl 0832.34035 · doi:10.1137/S0036139993253201
[11] Huo, H.F.: Periodic solutions for a semi-ratio-dependent predator-prey system with functional responses. Appl. Math. Lett. 18, 313--320 (2005) · Zbl 1079.34515 · doi:10.1016/j.aml.2004.07.021
[12] Ruan, S., Xiao, D.: Global analysis in a predator-prey system with nonmonotonic functional response. SIAM J. Appl. Math. 61, 1445--1472 (2001) · Zbl 0986.34045 · doi:10.1137/S0036139999361896
[13] Sokol, W., Howell, J.A.: Kinetics of phenol oxidation by washed cells. Biotechnol. Bioeng. 23, 2039--2049 (1980) · doi:10.1002/bit.260230909
[14] Wang, Q., Fan, M., Wang, K.: Dynamics of a class of nonautonomous semi-ratio-dependent predator-prey system with functional responses. J. Math. Anal. Appl. 278, 443--471 (2003) · Zbl 1029.34042 · doi:10.1016/S0022-247X(02)00718-7
[15] Xiao, D., Ruan, S.: Multiple bifurcations in a delayed predator-prey system with nonmonotonic functional response. J. Differ. Equ. 176, 494--510 (2001) · Zbl 1003.34064 · doi:10.1006/jdeq.2000.3982