The Pure and Applied Mathematics (한국수학교육학회지시리즈B:순수및응용수학)

Korean Society of Mathematical Education (한국수학교육학회)
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PHASE ANALYSIS FOR THE PREDATOR-PREY SYSTEMS WITH PREY DENSITY DEPENDENT RESPONSE

  • Received : 2018.10.08
  • Accepted : 2018.11.25
  • Published : 2018.11.30
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Abstract

This paper looks into phase plane behavior of the solution near the positive steady-state for the system with prey density dependent response functions. The positive invariance and boundedness property of the solution to the objective model are proved. The existence result of a positive steady-state and asymptotic analysis near the positive constant equilibrium for the objective system are of interest. The results of phase plane analysis for the system are proved by observing the asymptotic properties of the solutions. Also some numerical analysis results for the behaviors of the solutions in time are provided.

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SHGHCX_2018_v25n4_345_f0001.png 이미지

Figure 1. The graphs of the function u(t) and the function v(t) for the model (1) with α = 2, β = 1, γ = −0:5, and u(0) = 3, v(0) = 1

SHGHCX_2018_v25n4_345_f0002.png 이미지

Figure 2. The phase plane behavior of (u(t); v(t)) for the model (1) with α = 2, β = 1, γ = −0:5, and u(0) = 3, v(0) = 1

SHGHCX_2018_v25n4_345_f0003.png 이미지

Figure 3. The graphs of the function u(t) and the function v(t) for the model (1) with α = 4, β = 1, γ = −0:5, and u(0) = 3, v(0) = 1

SHGHCX_2018_v25n4_345_f0004.png 이미지

Figure 4. The phase plane behavior of (u(t); v(t)) for the model (1) with α = 4, β = 1, γ = −0:5, and u(0) = 3, v(0) = 1

References

  1. M.E. Gilpin: Do hares eat lynx?. American Naturalist. 107 (1973), 727-730. https://doi.org/10.1086/282870
  2. C. Holling: The components of predation as revealed by a study of small-mammal predation of the European pine sawfly. Can. Entomol. 91 (1959), 293-320. https://doi.org/10.4039/Ent91293-5
  3. C. Holling: The functional response of predators to prey density and its role in mimicry and population regulation. Mem. Entomol. Soc. Can. 45 (1965), 3-60.
  4. C. Jost, G. Devulder, J.A. Vucetich, R. Peterson & R. Arditi: The wolves of Isle Royale display scale-invariant satiation and density dependent predation on moose. J. Anim. Ecol. 74 (2005), no. 5, 809-816. https://doi.org/10.1111/j.1365-2656.2005.00977.x
  5. A.J. Lotka: Contribution to the Theory of Periodic Reaction. J. Phys. Chem. 14 (1910), no. 3, 271-274.
  6. A.J. Lotka: Analytical Note on Certain Rhythmic Relations in Organic Systems. Proc. Natl. Acad. Sci. U.S.A. 6 (1920), 410-415. https://doi.org/10.1073/pnas.6.7.410
  7. M.L. Rosenzweig & R.H. MacArthur: Graphical representation and stability conditions of predator-prey interactions. American Naturalist. 97 (1963), 209-223. https://doi.org/10.1086/282272
  8. S. Shim: Hopf Bifurcation Properties og Holling Type Predator-Prey System. J. Korea Soc. Math. Educ. Ser. B: Pure Appl. Math. 15 (2008), no. 3, 329-342.