Abstract
A predator-prey interaction is considered, where the prey has a stage structure — i.e., two life stages, immature and mature. The predator consumes both the immature and mature prey, and the prey is more prone to the predator at higher prey population densities. Both local and global stability of the system equilibria are discussed. With harvesting of the mature prey, there are threshold conditions for a sustainable yield.
Results in this paper were presented at the International Conference on Mathematical Modelling and Computation held at the University of Brunei Darussalam during 5–8 June 2006. in conjunction with the 20th anniversary celebration of the foundation of the university. This work was supported by SQU Grant IG/SCI/DOMS/03/05.
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References
C. R. Fisher-Piett, Soc. Biolgeogr, 92, 47–48 (1934).
J.H. Lawton, J. R. Beddington and R. Bonser, Switching in invertebrate predators. Ecological Studies, 9, 144–158 (1974).
W. W. Murdoch and A. Oaten, Predation and population stability. Adv. Ecol. Res., 9, 1–131 (1975).
C.S. Holling, Principles of insect predation. Ann. Rev. Entomol. 6, 163–182 (1961).
F. Takahashi, Reproduction curve with two equilibrium points: a consideration of the fluctuation of insect population. Res. Pop. Ecol. 47, 733–745 (1964).
R. M. May, Stability and Complexity in model ecosystems. Princeton, NJ: Princeton University Press (1973).
R. M. May, Some mathematical problems in biology, Providence, RI. Am. Math. Soc., 4, 11–29 (1974).
W. W. Murdoch and A. Oaten, Predation and population stability Adv. Ecol. Res., 9, 1–131 (1975).
J. Roughgarden and M. Feldman, Species packing and predation pressure, Ecology, 56, 489–492 (1975).
M. Tansky, Switching effects in prey-predator system, J. Theor. Biol. 70, 263–271 (1978).
Prajneshu and P. Holgate, A prey-predator model with switching effect. J. Theor. Biol. 125, 61-61–66 (1987).
Q. J. A. Khan, B. S. Bhatt and R. P. Jaju, Stability of a switching model with two habitats and a predator, J. Phys. Soc. Jpn., 63, 1995–2001 (1994).
Q. J. A. Khan, B. S. Bhatt and R. P. Jaju, Hopf Bifurcation analysis of a predator-prey system involving switching, J. Phys. Soc. Jpn., 65,3, 864–867 (1996).
M. Lloyd and H. S. Dybas, The periodical cicada problem, Evolution, 20, 133–149, 466–505 (1966).
E. D. LeCaven, C. Kipling, J. C. McCormack, A study of the numbers, biomass and year-class strengths of perch (perca fluviatillis L) in winteremiere, J. Anim., Ecol. 46, 281–306 (1977).
L. Nielsen, Effect of Walleye (Stizostedion Vitreun), Predation on Juvenile mortality and recruitment of yellow pereh (perca flavereens) in Oneida lake, New York. Can. J. Fish. Aquat. Sci. 37, 11–19 (1980).
H.I. Freedman, J.W.-H. So and J. Wu, A model for the growth of a population exhibiting stage structure: Cannibalism and cooperation, J. Comp. and App. Math., 52, 177–198 (1994).
W. S. C. Gurney, R. M. Nisbet and J. H. Lawton, The systematic formulation of tractable single-species population models incorporating age structure, J. Animal Ecol. 52, 479–495 (1983).
X. Song and L. Chen, Optimal harvesting and stability for a two-species competitive system with stage structure, Math. Biosci., 170, 173–186 (2001).
Q. J. A. Khan, E. V. Krishnan and M. A. Al-Lawatia, A stage structure model for the growth of a population involving switching and cooperation, Z. Angew. Math. Mech, 82, 125–135 (2002).
X. Zhang, L. Chen and A. U. Neumann, The stage structured predator-prey model and optimal harvesting policy, Math. Biosci, 168, 201–210 (2000).
W. Wang and L. Chen, A predator-prey system with stage structure for predator, Comput. Math. Appl. 33, 207 (1997).
C.W. Clark, Mathematical Bioeconomics: The Optimal management of renewable resources, 2nd Ed., Wiley, New York (1990).
A. W. Leng, Optimal harvesting-coefficient control of steady-state prey-predator diffusive Volterra-Lotka systems, Appl. Math. Optim, 31, 219 (1995).
D. K. Bhattacharya and S. Begun, Bioeconomic equilibrium of two-species systems I, Math. Biosci., 135, 111 (1996).
T.L. John, Variational Calculus and Optimal control, Springer, New York (1996).
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Khan, Q.J.A., Benkherouf, L., Smaoui, N. (2008). Stability and Optimal Harvesting in a Stage Structure Predator-Prey Switching Strategy. In: Hosking, R.J., Venturino, E. (eds) Aspects of Mathematical Modelling. Mathematics and Biosciences in Interaction. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8591-0_4
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