Breeding adjustment of small mammals to avoid predation

We studied theoretical models based on field and laboratories experiments where due to short-term predation pressure small mammals suppress breeding and when the predation pressure eases the suppressor class starts breeding again. The predator consumes both the breeder and suppressor individuals and this prey population is more prone to predation at higher densities. The stability analysis has been carried out for the equilibrium set for two models, in terms of the values of combinations of these parameters. We found that Hopf bifurcation will occur by varying a parameter q"1 which represents the rate by which breeder population turns suppressor population. It is found that predator induced breeding suppression (PIBS) acts to destabilize the stable interaction. We further examined the effect of time delay upon the stability of equilibrium in models. Using time delay as a bifurcation parameter it is shown that Hopf bifurcation could occur. We discussed these findings in the light of the Fennoscandian vole cycle. The theoretical results are compared with the numerical results for different sets of parameters.

[1]  W. Leontief.  Studies in the Structure of the American Economy: Theoretical and Empirical Explorations in Input-Output Analysis , 1953 .

[2]  J. Hale,et al.  Methods of Bifurcation Theory , 1996 .

[3]  S. Levin.  Lectu re Notes in Biomathematics , 1983 .

[4]  Richard Bellman,et al.  Differential-Difference Equations , 1967 .

[5]  M. Kalecki,et al.  A Macrodynamic Theory of Business Cycles , 1935 .

[6]  R. D. Driver,et al.  Ordinary and Delay Differential Equations , 1977 .

[7]  Giancarlo Gandolfo,et al.  Mathematical methods and models in economic dynamics , 1971 .

[8]  O. Mustonen,et al.  Influence of predation risk on early development and maturation in three species of Clethrionomys voles , 1993 .

[9]  K. Gopalsamy.  Stability and Oscillations in Delay Differential Equations of Population Dynamics , 1992 .

[10]  Hanna Kokko,et al.  BREEDING SUPPRESSION AND PREDATOR–PREY DYNAMICS , 2000 .

[11]  G. Sell,et al.  The Hopf Bifurcation and Its Applications , 1976 .

[12]  H. Ylönen,et al.  Behaviour of cyclic bank voles under risk of mustelid predation: do females avoid copulations? , 1994, Oecologia.

[13]  H. Ylönen,et al.  Antipredatory behaviour of Clethrionomys voles―David and Goliath' arms race , 1992 .

[14]  Lauri Oksanen,et al.  Optimization of reproductive effort and foraging time in mammals: The influence of resource level and predation risk , 2005, Evolutionary Ecology.

[15]  J. Craggs.  Applied Mathematical Sciences , 1973 .

[16]  J. Valkama,et al.  Reproductive investment under fluctuating predation risk: Microtine rodents and small mustelids , 1994, Evolutionary Ecology.

[17]  N. G. Parke,et al.  Ordinary Differential Equations. , 1958 .

[18]  M. Hasse,et al.  W. H. Greub, Multilinear Algebra. (Die Grundlehren der mathematischen Wissenschaften. Band 136). XII + 225 S. Berlin/Heidelberg/New York 1967. Springer-Verlag. Preis geb. DM 32,– , 1971 .

[19]  I. Hanski,et al.  A predator-prey model with optimal suppression of reproduction in the prey. , 1996, Mathematical biosciences.

[20]  V. Arnold,et al.  Ordinary Differential Equations , 1973 .

[21]  Jack K. Hale,et al.  Introduction to Functional Differential Equations , 1993, Applied Mathematical Sciences.

[22]  H. Ylönen.  Vole cycles and antipredatory behaviour. , 1994, Trends in ecology & evolution.

[23]  H. Ylönen,et al.  Breeding suppression in the bank vole as antipredatory adaptation in a predictable environment , 1994, Evolutionary Ecology.

[24]  E. Koskela,et al.  Suppressed breeding in the field vole (Microtus agrestis): an adaptation to cyclically fluctuating predation risk , 1995 .

[25]  S. L. Lima,et al.  Predator–induced breeding suppression and its consequences for predator–prey population dynamics , 1997, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[26]  D. Hik.  Does risk of predation influence population dynamics? Evidence from cyclic decline of snowshoe hares , 1995 .

[27]  N. Macdonald.  Time lags in biological models , 1978 .

[28]  H. Ylönen.  Weasels Mustela Nivalis Suppress Reproduction in Cyclic Bank Voles Clethrionomys Glareolus , 1989 .

[29]  S. L. Lima,et al.  Behavioral decisions made under the risk of predation: a review and prospectus , 1990 .