Discrete-time analogues of predator-prey models with monotonic or nonmonotonic functional responses

Nonlinear Analysis: Real World Applications

Volumn 8, Issue 4, 2007, Pages 1079-1095

  Xia, Yonghui ^a   Cao, Jinde ^b   Lin, Muren ^a  

^a FUZHOU UNIVERSITY   (China)

^b SOUTHEAST UNIVERSITY   (China)

Author keywords

Coincidence degree; Discrete time analogues; Periodic solutions; Predator prey model

Indexed keywords

DISCRETE TIME CONTROL SYSTEMS; FUNCTION EVALUATION; NUMERICAL METHODS; REAL TIME CONTROL;

COINCIDENCE DEGREE; DISCRETE TIME ANALOGUES; PERIODIC SOLUTIONS; PREDATOR PREY MODELS;

MATHEMATICAL MODELS;

EID: 34047261092     PISSN: 14681218     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.nonrwa.2006.06.007     Document Type: Article

References (46)

1. Agarwal R.P. Difference Equations and Inequalities: Theory, Method and Applications Monographs and Textbooks in Pure and Applied Mathematics vol. 228 (2000), Marcel Dekker, New York
2. Agarwal R.P., and Wong P.J.Y. Advance Topics in Difference Equations (1997), Kluwer Academic Publishers, Dordrecht
3. Andrews J.F. A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates. Biotechnol. Bioeng. 10 (1986) 707-732
4. Arditi R., and Ginzburg L.R. Coupling in predator-prey dynamics: ratio-dependence. J. Theoret. Biol. 139 (1989) 311-326
5. Berryman A.A. The origins and evolution of predator-prey theory. Ecology 75 (1992) 1530-1535
6. Bush A.W., and Cook A.E. The effect of time delay and growth rate inhibition in the bacterial treatment of wastewater. J. Theoret. Biol. 63 (1976) 385-395
7. Caperson J. Time lag in population growth response of isochrysis galbana to a variable nitrate environment. Ecology 50 (1969) 188-192
8. Chen L. Mathematical Models and Methods in Ecology (1998), Science Press, Beijing (in Chinese)
9. Fan M., and Wang K. Optimal harvesting policy for single population with periodic coefficients. Math. Biosci. 152 2 (1998) 165-177
10. Fan M., and Wang K. Periodicity in a delayed ratio-dependent predator-prey system. J. Math. Anal. Appl. 262 (2001) 179-190
11. Fan M., and Wang K. Periodic solutions of a discrete time nonautonomous ratio-dependent predator-prey system. Math. Comput. Modelling 35 9-10 (2002) 951-961
12. Fan M., and Wang Q. Periodic solutions of a class of nonautonomous discrete time semi-ratio-dependent predator-prey systems. Discrete Continuous Dyn. Syst. B (DCDS-B) 4 (2004) 563-574
13. Fan M., Wang Q., and Zou X.F. Dynamics of a nonautonomous ratio-dependent predator-prey system. Proc. R. Soc. Edinburgh Sect. A 133 (2003) 97-118
14. Fan Y., Li W., and Wang L. Periodic solutions of delayed ratio-dependent predator-prey models with monotonic or nonmonotonic functional responses. Nonlinear Anal. Real World Appl. 5 (2004) 247-263
15. Frddeman H.I. Deterministic Mathematical Models in Population, Ecology (1980), Marcel Dekker, New York
16. Freedman H.I., and Kuang Y. Stability switches in linear scalar neutral delay equations. Funkc. Ekvac. 34 (1991) 187-209
17. Freedman H.I., and Mathsen R.M. Persistence in predator-prey systems with ratio-dependent predator influence. Bull. Math. Biol. 55 (1993) 817-827
18. Gaines R.E., and Mawhin J.L. Coincidence Degree and Nonlinear Differential Equations (1977), Springer, Berlin
19. Goh B.S. Management and Analysis of Biological Populations (1980), Elsevier, The Netherlands
20. Gopalsamy K. Global asymptotic stability in Volterra's population systems. J. Math. Biol. 19 (1984) 157-168
21. Gopalsamy K. Stability and oscillation in delay differential equations of population dynamics. Mathematics and its Applications vol. 74 (1992), Kluwer Academic Publishers, Dordrecht
22. Gopalsamy K., He X., and Wen L. On a periodic neutral logistic equation. Glasgow Math. J. 33 (1991) 281-286
23. Gopalsamy K., and Zhang B.G. On a neutral delay logistic equation. Dyn. Stab. Syst. 2 (1998) 183-195
24. Hsu S.B., Hwang T.W., and Kuang Y. Global analysis of Michaelis-Menten type ratio-dependent predator-prey system. J. Math. Biol. 42 (2003) 489-506
25. Jost C., Arino O., and Arditi R. About deterministic extinction in ratio-dependent predator-prey models. Bull. Math. Biol. 61 (1999) 19-32
26. Kuang Y. Delay Differential Equations with Application Dynamic (1988), Academic Press, New York
27. Kuang Y. Rich dynamics of Gause-type ratio-dependent predator-prey systems. Fields Inst. Commun. 21 (1999) 325-337
28. Kuang Y., and Beretta E. Global qualitative analysis of a ratio-dependent predator-prey systems. J. Math. Biol. 36 (1998) 389-406
29. Lakshmikantham V., Bainov D.D., and Simeonov P.S. Theory of Impulsive Differential Equations (1989), World Scientific, Singapore
30. Liu X., and Ballinger G. Uniform asymptotic stability of impulsive delay differential equations. Comput. Math. Appl. 41 (2001) 903-915
31. Liu X., and Ballinger G. Existence and continuability of solutions for differential equations with delays and state-dependent impulses. Nonlinear Anal. 51 (2002) 633-647
32. Liu X., and Ballinger G. Boundedness for impulsive delay differential equations and applications to population growth models. Nonlinear Anal. 53 (2003) 1041-1062
33. Lundberg P., and Fryxell J.M. Expected population density versus productivity in ratio-dependent and prey-dependent models. Am. Nat. 147 (1995) 153-161
34. Murry J.D. Mathematical Biology (1989), Springer, New York
35. Ruan S., and Xiao D. Global analysis in a predator-prey system with nonmonotonic functional response. SIAM J. Appl. Math. 61 (2001) 1445-1472
36. Sokol W., and Howell J.A. Kinetics of phenol oxidation by washed cells. Biotechnol. Bioeng. 23 (1980) 2039-2049
37. Wang L.L., and Li W.T. Existence and global stability of positive periodic solutions of a predator-prey system with delays. Appl. Math. Comput. 146 (2003) 167-185
38. Wang Q., Fan M., and Wang K. Dynamics of a class of nonautonomous semi-ratio-dependent predator-prey systems with functional responses. J. Math. Anal. Appl. 278 2 (2003) 443-471
39. Y.H. Xia, Positive Periodic Solution for a Neutral Impulsive Delayed Lotka-Volterra Competition Systems with the Effect of Toxic Substance, Nonlinear Anal.: Real World Appl., in press, doi:10.1016/j.nonrwa.2005.07.002.
40. Xia Y.H., and Cao J. Almost periodicity in an ecological model with M-predators and N-preys by "pure-delay type" system. Nonlinear Dyn. 39 3 (2005) 275-304
41. Y.H. Xia, J. Cao, M. Lin, New results on the existence and uniqueness of almost periodic solutions for BAM neural networks with continuously distributed delays, Chaos Solitons Fractals, in press, doi:10.1016/j.chaos.2005.10.043.
42. Xia Y.H., Chen F., Chen A., and Cao J. Existence and global attractivity of an almost periodic ecological model. Appl. Math. Comput. 157 2 (2004) 449-475
43. Xia Y.H., Lin M., and Cao J. The existence of almost periodic solutions of certain perturbation system. J. Math. Anal. Appl. 310 1 (2005) 81-96
44. Xiao D., and Ruan S. Multiple bifurcations in a delayed predator-prey system with nonmonotonic functional response. J. Differential Equations 176 (2001) 494-510
45. Xiao D.M., and Ruan S.G. Global dynamics of a ratio-dependent predator-prey system. J. Math. Biol. 43 3 (2001) 268-290
46. Zhang R.Y., Wang Z.C., Chen Y., and Wu J. Periodic solutions of a single species discrete population model with periodic harvest/stock. Comput. Math. Appl. 39 (2000) 77-90