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1. 北京航空航天大学 数学与系统科学学院, 北京 100083;
2. 肯特大学 数学、统计与精算学院, 坎特伯雷 CT27NZ;
3. 北京大学 数学科学学院, 北京 100871

Qualitative analysis of a food chain model with alternative prey
LI Ya1 , HU Hanyan2 , LIN Junyi3
1. School of Mathematics and Systems Science, Beijing University of Aeronautics and Astronautics, Beijing 100083, China ;
2. School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury CT27NZ, United Kingdom ;
3. School of Mathematics, Peking University, Beijing 100871, China
Received: 2015-09-28; Accepted: 2015-12-18; Published online: 2016-02-18 10:54
Foundation item: the Fundamental Research Funds for the Central Universities (30361201, 30458701)
Corresponding author. Tel.:15801211607, E-mail:yli@buaa.edu.cn
Abstract: Studies in ecology show that alternative prey has important effects on predator-prey system in the aspects of system stability, species persistence, etc. For species at the intermediate level of a food chain, they are prey of top predators, while predating the bottom prey at the same time. Therefore, the effect of changes in intermediate predators' population on the system cannot be ignored. We set up a three-dimensional food chain model composed of top predator, intermediate predator and prey, assuming that the intermediate predator had alternative choice of prey. The existence and stability of the equilibria were derived through dynamical study of the system. The existence of Hopf bifurcation was explored in depth. The numerical simulations used the prey choice parameter as the bifurcation variable to study the possible Hopf bifurcations. The effect of alternative prey on the stability of the food chain system was derived through the theoretical and numerical results.
Key words: predator-prey model     food chain     alternative prey     locally asymptotically stable     Hopf bifurcation

1 模型的建立

 (1)

 (2)

 (3)

2 平衡点的存在性和稳定性 2.1 边界平衡点的存在性和稳定性分析

1) (0，0，0)，即3种群数量均为0。此时模型在平衡点处的雅可比行列式为

2) (1, 0, 0)，即食饵数量达到最大值，中间捕食者和顶级捕食者种群数量为零，雅可比行列式为

3) (x, y, 0)，即只有顶级捕食者数量为0，而中间捕食者和食饵数量非0。由方程组

 (4)
 (5)

，则

g(x)x>0时单调递增，从而有，即, 此时平衡点存在。

2.2 内部平衡点的存在性和稳定性分析

2.2.1 内部平衡点(x*, y*, z*)的存在性

+c(1-A)-d1>0时，(x*, y*, z*)为系统式(1)的一个内部平衡点。

=0得到内部平衡点处的x满足

 图 1 内部平衡点的存在性 Fig. 1 Existence of interior equilibrium point

2.2.2 内部平衡点(x*, y*, z*)的稳定性

P1P2-P3进行化简：

 (6)

 图 2 抛物线f(s)的图像 Fig. 2 Graph of parabola f(s)

3 Hopf分支的数值仿真

c < c*时，内部平衡点局部渐进稳定，而当c=c*时，出现Hopf分支。图 3(a)是利用软件XPPAUT绘制出的以c为参数的分支图，其他参数取值为：a1=6, a2=0.2, b1=2, b2=0.05, d1=0.3, d2=0.04，A=0.2。

c*≈1.5时产生Hopf分支。在此点左侧，内部平衡点局部渐进稳定(图 3(b))；在此点右侧，内部平衡点变为不稳定，出现局部渐进稳定的周期解(图 3(c))。

 图 3 c为分支参数时的系统稳定性 Fig. 3 Stability of system with bifurcation parameter c

 图 4 A为分支参数时的系统稳定性 Fig. 4 Stability of system with bifurcation parameter A

4 结论

1) 种群对食饵选择性的不同，会引起边界平衡点稳定性的改变。中间捕食者增加对其他可供选择的食饵的捕食比率，有利于其生存。

2) 捕食者对其他食饵的选择性也对系统的3个种群共存有着不可忽略的影响。当选择参数A较小或者对其他食饵的摄食量较多时，则系统中某些种群可能消失；而随着中间捕食者对其他食饵的捕食量的增大，系统中的3个种群将以周期振荡的形式共存于系统或者最终数量达到稳定。

 [1] FREEDMAN H I, WALTMAN P. Mathematical analysis of some three-species food chain models[J]. Mathematical Biosciences, 1977, 33 (3) : 257 –276. [2] HASTINGS A, POWELL T. Chaos in a three-species food chain[J]. Ecology, 1991, 72 (3) : 896 –903. DOI:10.2307/1940591 [3] HSU S, HWANG T, KUANG Y. A ratio-dependent food chain model and its applications to biological control[J]. Mathematical Biosciences, 2003, 181 (1) : 55 –83. DOI:10.1016/S0025-5564(02)00127-X [4] GINOUX J M, ROSSETTO B, JAMET J L. Chaos in a three-dimensional Volterra-Gause model of predator-prey type[J]. International Journal of Bifurcation and Chaos, 2005, 15 (5) : 1689 –1708. DOI:10.1142/S0218127405012934 [5] PATHAK S, MAITI A, SAMANTA G P. Rich dynamics of a food chain model with Hassell-Varley type functional responses[J]. Applied Mathematics and Computation, 2009, 208 (2) : 303 –317. DOI:10.1016/j.amc.2008.12.015 [6] GEORGE A K, KOOI P I, BOER M P. Ecological consequences of global bifurcations in some food chain models[J]. Mathematical Biosciences, 2010, 226 (2) : 120 –133. DOI:10.1016/j.mbs.2010.04.005 [7] STRAUSS S Y. Indirect effects in community ecology: Their definition, study and importance[J]. Trends in Ecological Evolution, 1991, 6 (7) : 206 –210. DOI:10.1016/0169-5347(91)90023-Q [8] SAHOO B, PORIA S. Diseased prey predator model with general Holling type interactions[J]. Applied Mathematics and Computation, 2014, 226 (1) : 83 –100. [9] HUXEL G R, MCCANN K. Food web stability: The influence of trophic flows across habitats[J]. American Natualist, 1998, 152 (3) : 460 –469. DOI:10.1086/286182 [10] HUXEL G R, MCCANN K, POLIS G A. Effects of partitioning allochthonous and autochthonous resources on food web stability[J]. Ecological Research, 2002, 17 (4) : 419 –432. DOI:10.1046/j.1440-1703.2002.00501.x [11] SRINIVASU P D N, PRASAD B S R V, VENKATESULU M. Biological control through provision of additional food to predators: A theoretical study[J]. Theoretical Population Biology, 2007, 72 (1) : 111 –120. DOI:10.1016/j.tpb.2007.03.011 [12] SAHOO B.Effects of additional foods to predators on nutrient-consumer-predator food chain model[J/OL].ISRN Biomathematics, 2012.http://dx.doi.org/10.5402/2012/796783. [13] SAHOO B.Global stability of predator-prey system with alternative prey[J/OL].ISRN biotechnology, 2013.http://dx.doi.org/10.5402/2013/898039. [14] SAHOO B, PORIA S. Disease control in a food chain model supplying alternative food[J]. Applied Mathematical Modelling, 2013, 37 (8) : 5653 –5663. DOI:10.1016/j.apm.2012.11.017 [15] SAHOO B, PORIA S. The chaos and control of a food chain model supplying additional food to top-predator[J]. Chaos Solitons & Fractals, 2014, 58 (1) : 52 –64.

#### 文章信息

LI Ya, HU Hanyan, LIN Junyi

Qualitative analysis of a food chain model with alternative prey

Journal of Beijing University of Aeronautics and Astronsutics, 2016, 42(10): 2075-2081
http://dx.doi.org/10.13700/j.bh.1001-5965.2015.0642