Choose the values of parameters as in Table II and calculate the values λ1 =
a1d1
e1m1−d1 = 2.4 and λ2 = e a2d2
2m2−d2 = 4.5. Now, using K (the carrying capacity of the resource) as a bifurcation parameter, increase K from 4.5 to 80 and calculate fx∗(K) as a function of K in (2.22). We can see that fx∗ is monotonically increasing from negative to positive (see the first graph of Fig. I). The values of functions α1, α3, and α1α2 − α3 are also calculated (see the 2nd - 4th graphs of Fig. I).
The dynamics of solutions with respect to the capacity K are illustrated in Figure II(a)-(e).
(i) 0 < K = 2 < λ1. The semi-trivial equilibrium EK is globally asymptotically stable, (see Fig. II:(a))
0 10 20 30 40 50 60 70 80
−1.5
−1.0
−0.5 0.0 0.5
fx
0 10 20 30 40 50 60 70 80
0.00.2 0.40.6 0.81.0
a1
0 10 20 30 40 50 60 70 80
−0.02
−0.010.000.010.020.030.04
a3
0 10 20 30 40 50 60 70 80
K
−0.050.000.050.100.150.200.250.30
a1a2-a3
Figure I: The graphs of fx∗(K), α1(K), α3(K) and α1(K)α2(K) − α3(K) in terms of K as K increases from 4.5 to 80.
(ii) λ1 < K = 3 < λ2. The semi-trivial equilibrium E1 is globally asymptotically stable, (see Fig. II:(b))
(iii) λ2 < K = 10. The solution converges to the positive equilibrium Ec as t → ∞. We can see that the positive equilibrium is asymptotically stable, (see Fig. II:(c))
(iv) K = 75. The positive equilibrium Ec loses its stability and a periodic solution bifurcates from it. (see Fig. II:(d))
Next, we do some numerical simulations of system (1.1) with interference ef-fects, i.e., b1 6= 0 and b2 6= 0. In order to compare the differences of solutions of system (1.1) with or without interference effects, we choose the same parameters as those in Fig. 3 of [11] in Table III. We plot limit cycles of population of predator 1 against that of predator 2 in Fig III. Fig III (a) is for b1 = 0, b2 = 0, (b) is for
Table III: Parameter Values for the Case with Interference.
r = 20 · ln 2 a1 = 200 d1 = ln 2/2 e1 = 0.1 m1 = 10 · ln 2 K = 1100 a2 = 500 d2 = ln 2 e2 = 1.4 m2 = 2 · ln 2 .
b1 = 0, b2 = 1, and (c) is for b1 = 1, b2 = 0. All above three limit cycles are plotted in a graph showed in (d). With the same parameters, we compute the numerical solutions of (1.1) with various parameters b1 and b2. In Fig III (e) and (f) show the numerical results where b1, b2 are varied from 0 to 10 with step-size 0.1 in (e) and b1, b2 are varied from 0 to 1 with step-size 0.01 in (e). The white region represents that the solutions are periodic and the black region means that the solutions approach a positive equilibrium.
7 Discussion
In systems where there is no intraspecific or interspecific interference but only ex-ploitative competition for resources between consumers, the conventional wisdom is that the number of consumer species which can coexist is less than or equal to the number of distinct resources. This has been shown in a number of mod-els including chemostat modmod-els (Smith and Waltman [21]). In the case of two consumers, our results indicate that sufficiently strong intraspecific and mutual interference can ensure the long term survival of both competing consumers not only in terms of uniform persistence but also in the sense of global stability. A possible explanation of this phenomenon is that intraspecific feeding interference in one consumer reduces its equilibrium density and allows the other consumer to have better access to the resource. It is also interesting to observe that when the carrying capacity K of the resource is increased, the interior equilibrium loses stability and a three dimensional positive periodic solution arises via Hopf bifur-cation. This indicates that the paradox of enrichment phenomenon may occur for the two-predators-one-prey model as well.
References
[1] R. A. Armstrong and R. McGehee. Competitive exclusion. American Natu-ralist, 115(2):151–170, 1980.
[2] J. R. Beddington. Mutual Interference Between Parasites or Predators and its Effect on Searching Efficiency. The Journal of Animal Ecology, 44(1):331, Feb. 1975.
[3] G. Butler. Persistence in dynamical systems. J Differential Equations, 1986.
[4] G. Butler, H. I. Freedman, and P. Waltman. Uniformly persistent systems.
Proceedings of the American Mathematical Society, 96(3):425–430, 1986.
[5] G. J. Butler and P. Waltman. Bifurcation from a limit cycle in a two predator-one prey ecosystem modeled on a chemostat. Journal of Mathematical Biology, 12(3):295–310, 1981.
[6] R. S. Cantrell and C. Cosner. On the dynamics of predator-prey models with the Beddington-DeAngelis functional response. Journal of Mathematical Analysis and Applications, 257(1):206–222, 2001.
[7] D. L. DeAngelis, R. A. Goldstein, and R. V. O’neill. A model for tropic interaction. Ecology, 56(4):881–892, 1975.
[8] H. I. Freedman, S. Ruan, and M. Tang. Uniform Persistence and Flows Near a Closed Positively Invariant Set. Journal of Dynamics and Differential Equations, 6(4):583–600, 1994.
[9] S. B. Hsu. Limiting Behavior for Competing Species. SIAM Journal on Applied Mathematics, 34(4):760–763, June 1978.
[10] S. B. Hsu. A survey of constructing Lyapunov functions for mathematical models in population biology. Taiwanese Journal of Mathematics, 9(2):151–
173, 2005.
[11] S. B. Hsu, S. P. Hubbell, and P. Waltman. A contribution to the theory of competing predators. Ecological Monographs, 48(3):337–349, 1978.
[12] S. B. Hsu, S. P. Hubbell, and P. Waltman. Competing predators. SIAM Journal on Applied Mathematics, 35(4):617–625, 1978.
[13] G. Huisman and R. J. DeBoer. A formal derivation of the ”Beddington”
functional response. Journal of Theoretical Biology, 185(3):389–400, 1997.
[14] T.-W. Hwang. Global analysis of the predator-prey system with Beddington-DeAngelis functional response. Journal of Mathematical Analysis and Appli-cations, 281(1):395–401, 2003.
[15] T.-W. Hwang. Uniqueness of limit cycles of the predator-prey system with Beddington-DeAngelis functional response. Journal of Mathematical Analysis and Applications, 290(1):113–122, 2004.
[16] J. P. Keener. Oscillatory coexistence in the chemostat: a codimension two unfolding. SIAM Journal on Applied Mathematics, 43(5):1005–1018, 1983.
[17] W. Liu, D. Xiao, and Y. Yi. Relaxation oscillations in a class of predator-prey systems. Journal of Differential Equations, 188(1):306–331, 2003.
[18] S. Muratori and S. Rinaldi. Remarks on competitive coexistence. SIAM Journal on Applied Mathematics, 49(5):1462–1472, 1989.
[19] H. L. Smith. The interaction of steady state and Hopf bifurcations in a two-predator-one-prey competition model. SIAM Journal on Applied Mathe-matics, 42(1):27–43, 1982.
[20] H. L. Smith and H. R. Thieme. Dynamical systems and population persistence, volume 118 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2011.
[21] H. L. Smith and P. Waltman. The Theory of the Chemostat, volume 13 of Cambridge Studies in Mathematical Biology. Cambridge University Press, Cambridge, 1995.
0 50 100 150 200 asymptotically stable. In Fig II (d), K = 75, the periodic solution exist. Hopf bifurcation occurs between K = 70 and K = 75.
180 200 220 240 260 280 300 320 340 360
100 200 300 400 500 600 700 800 900
0
Figure III: The parameters are given in Table III. The graphs in Fig III (a), (b), (c) are the limit cycle solutions of system (1.1) projected in (y1, y2)-plane with b1 = b2 = 0 in Fig III(a), b1 = 0, b2 = 1 in Fig III (b), b1 = 1, b2 = 0 in Fig III (c). We put Fig III (a), (b), (c) in the same graph in Fig III (d). In Fig III (e), in the b1-B2 parameter region, 0 ≤ b1, b2 ≤ 1, the white region represents that the numerical solutions are periodic and the black region represents that the numerical
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