# American Institute of Mathematical Sciences

## Dynamics of a predator-prey model with state-dependent carrying capacity

 1 Department of Applied Mathematics, Yuncheng University, Yuncheng, Shanxi 044000, China 2 Department of Mathematics and Statistics, University of New Brunswick, Fredericton, NB E3B5A3, Canada 3 North-West Plateau Institute of Biology, Key Laboratory of Ecology Restoration in Cold Region in Qinghai Province, the Chinese Academy of Sciences, Xining, Qinghai 810001, China

* Corresponding author

Received  September 2017 Revised  May 2018 Published  February 2019

Vegetation and plateau pika are two key species in alpine meadow ecosystems on the Tibetan Plateau. It is frequently observed on the field that plateau pika reduces the carrying capacity of vegetation and the mortality of plateau pika increases along with the increasing height of vegetation. This motivates us to propose and study a predator-prey model with state-dependent carrying capacity. Theoretical analysis and numerical simulations show that the model exhibits complex dynamics including the occurrence of saddle-node bifurcation, transcritical bifurcation and Hopf bifurcation, and the coexistence of two stable equilibria.

Citation: Hanwu Liu, Lin Wang, Fengqin Zhang, Qiuying Li, Huakun Zhou. Dynamics of a predator-prey model with state-dependent carrying capacity. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019028
##### References:
 [1] T. Akiyama and K. Kawamura, Grassland degradation in China: Methods of monitoring management and restoration, Grassl. Sci., 53 (2007), 1-7. doi: 10.1111/j.1744-697X.2007.00073.x. [2] S. Cai and Z. Qian, Introduction of Qualitative Theory of Ordinary Differential Equations, Higher Education Press, Beijing, 1994. [3] A. Hastings and L. Gross, Encyclopedia of Theoretical Ecology, University of California Press, Berkeley, 2012. [4] C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, Mem. Entomol. Soc. Can., 97 (1965), 5-60. doi: 10.4039/entm9745fv. [5] G. E. Hutchinson, Circular causal systems in ecology, Ann. N.Y. Acad. Sci., 50 (1948), 221-246. doi: 10.1111/j.1749-6632.1948.tb39854.x. [6] C. Lai and A. Smith, Keystone status of plateau pika (Ochotona curzoniae): Effect of control on biodiversity of native birds, Biodivers. Conserv., 12 (2003), 1901-1912. [7] Q. Li, J. Li and Q. Dong, Study on the effect of fencing on Kobrecia parva alpine meadow pasture at different degrading stages in Yangtze and Yellow river headwaters, Pratacultural Sci., 23 (2006), 16-21. [8] Y. Li, X. Zhao and G. Cao, Analyses on climates and vegetation productivity background at Haibei alpine meadow ecosystem research station, Plateau Meteorol, 23 (2004), 558-567. [9] W. Liu, X. Wang, L. Zhou and H. Zhou, Studies on destruction, prevention and control of plateau pikas in Kobresia pygmaea meadow, Acta Theriol. Sin., 23 (2003), 214-219. [10] Z. Ma and Y. Zhou, The Qualitative and Stability Methods of Ordinary Differential Equations, Science Press, Beijing, 2001. [11] J. Qu, M. Liu and M. Yang, Reproduction of plateau pika (Ochotona curzoniae) on the Qinghai-Tibetan plateau, Eur. J. Wildl. Res., 58 (2012), 269-277. [12] M. L. Rosenzweig and R. H. Macarthur, Graphical representation and stability conditions of predator-prey interactions, Am. Nat., 97 (1963), 209-223. [13] S. Shen and Y. Chen, Preliminary research on ecology of the plateau pica at Dawu area, Guoluo, Qinghai province, Acta Theriol. Sin., 4 (1984), 107-115. [14] Y. Shi, On the influences of range land vegetation to the density of plateau pika (Ochotona curzoniae), Acta Theriol. Sin., 3 (1983), 181-187. [15] A. T. Smith and J. M. Foggin, The plateau pika (Ochotona curzoniae) is a keystone species for biodiversity on the Tibetan plateau, Anim. Conserv., 2 (1999), 235-240. [16] J. Wang, W. Wei and Y. Zhang, The sex ratio of plateau pika's population, Acta Theriol. Sin., 24 (2004), 177-181. [17] X. Wang and K. Dai, Natural longevity of plateau pika (Ochotona curzoniae), Acta Theriol. Sin., 9 (1989), 56-62. [18] X. Wang and A. T. Smith, On the natural winter mortality of the plateau pika (Ochotona Curzoniae), Acta Theriol. Sin., 8 (1988), 152-156. [19] X. Wang and X. Fu, Sustainable management of alpine meadows on the Tibetan Plateau: Problems overlooked and suggestions for change, Ambio, 33 (2004), 169-171. [20] X. Wei, S. Li, P. Yang and H. Cheng, Soil erosion and vegetation succession in alpine Kobresia steppe meadow caused by plateau pika-A case study of Nagqu County, Tibet, Chinese Geographical Science, 17 (2007), 75-81. [21] L. Wen, S. Dong and Y. Li, Effect of degradation intensity on grassland ecosystem services in the alpine region of Qinghai-Tibetan plateau, China, Plos One, 8 (2013), 1-7. doi: 10.1371/journal.pone.0058432. [22] Z. Yang and X. Jiang, The harm of plateau pika on grassland vegetation and its control threshold value, Pratacultural Sci., 19 (2002), 63-65. [23] B. Yin, J. Wang and W. Wei, Population reproductive characteristics of plateau pika in alpine meadow ecosystem, Acta Theriol. Sin., 24 (2004), 222-228. [24] Y. Zhang, X. Yuan and D. Niu, Response of plateau pika burrow density to vegetation management in an alpine meadow, Maqu County, Gansu, Acta Prataculturae Sin., 25 (2016), 87-94. [25] X. Zhao and X. Zhou, Ecological basis of alpine meadow ecosystem management in Tibet: Haibei alpine meadow ecosystem research station, Ambio, 28 (1999), 642-647. [26] H. Zhou, L. Zhou and X. Zhao, Stability of alpine meadow ecosystem on the Qinghai-Tibetan Plateau, Chin. Sci. Bull., 51 (2006), 320-327. doi: 10.1007/s11434-006-0320-4.

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##### References:
 [1] T. Akiyama and K. Kawamura, Grassland degradation in China: Methods of monitoring management and restoration, Grassl. Sci., 53 (2007), 1-7. doi: 10.1111/j.1744-697X.2007.00073.x. [2] S. Cai and Z. Qian, Introduction of Qualitative Theory of Ordinary Differential Equations, Higher Education Press, Beijing, 1994. [3] A. Hastings and L. Gross, Encyclopedia of Theoretical Ecology, University of California Press, Berkeley, 2012. [4] C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, Mem. Entomol. Soc. Can., 97 (1965), 5-60. doi: 10.4039/entm9745fv. [5] G. E. Hutchinson, Circular causal systems in ecology, Ann. N.Y. Acad. Sci., 50 (1948), 221-246. doi: 10.1111/j.1749-6632.1948.tb39854.x. [6] C. Lai and A. Smith, Keystone status of plateau pika (Ochotona curzoniae): Effect of control on biodiversity of native birds, Biodivers. Conserv., 12 (2003), 1901-1912. [7] Q. Li, J. Li and Q. Dong, Study on the effect of fencing on Kobrecia parva alpine meadow pasture at different degrading stages in Yangtze and Yellow river headwaters, Pratacultural Sci., 23 (2006), 16-21. [8] Y. Li, X. Zhao and G. Cao, Analyses on climates and vegetation productivity background at Haibei alpine meadow ecosystem research station, Plateau Meteorol, 23 (2004), 558-567. [9] W. Liu, X. Wang, L. Zhou and H. Zhou, Studies on destruction, prevention and control of plateau pikas in Kobresia pygmaea meadow, Acta Theriol. Sin., 23 (2003), 214-219. [10] Z. Ma and Y. Zhou, The Qualitative and Stability Methods of Ordinary Differential Equations, Science Press, Beijing, 2001. [11] J. Qu, M. Liu and M. Yang, Reproduction of plateau pika (Ochotona curzoniae) on the Qinghai-Tibetan plateau, Eur. J. Wildl. Res., 58 (2012), 269-277. [12] M. L. Rosenzweig and R. H. Macarthur, Graphical representation and stability conditions of predator-prey interactions, Am. Nat., 97 (1963), 209-223. [13] S. Shen and Y. Chen, Preliminary research on ecology of the plateau pica at Dawu area, Guoluo, Qinghai province, Acta Theriol. Sin., 4 (1984), 107-115. [14] Y. Shi, On the influences of range land vegetation to the density of plateau pika (Ochotona curzoniae), Acta Theriol. Sin., 3 (1983), 181-187. [15] A. T. Smith and J. M. Foggin, The plateau pika (Ochotona curzoniae) is a keystone species for biodiversity on the Tibetan plateau, Anim. Conserv., 2 (1999), 235-240. [16] J. Wang, W. Wei and Y. Zhang, The sex ratio of plateau pika's population, Acta Theriol. Sin., 24 (2004), 177-181. [17] X. Wang and K. Dai, Natural longevity of plateau pika (Ochotona curzoniae), Acta Theriol. Sin., 9 (1989), 56-62. [18] X. Wang and A. T. Smith, On the natural winter mortality of the plateau pika (Ochotona Curzoniae), Acta Theriol. Sin., 8 (1988), 152-156. [19] X. Wang and X. Fu, Sustainable management of alpine meadows on the Tibetan Plateau: Problems overlooked and suggestions for change, Ambio, 33 (2004), 169-171. [20] X. Wei, S. Li, P. Yang and H. Cheng, Soil erosion and vegetation succession in alpine Kobresia steppe meadow caused by plateau pika-A case study of Nagqu County, Tibet, Chinese Geographical Science, 17 (2007), 75-81. [21] L. Wen, S. Dong and Y. Li, Effect of degradation intensity on grassland ecosystem services in the alpine region of Qinghai-Tibetan plateau, China, Plos One, 8 (2013), 1-7. doi: 10.1371/journal.pone.0058432. [22] Z. Yang and X. Jiang, The harm of plateau pika on grassland vegetation and its control threshold value, Pratacultural Sci., 19 (2002), 63-65. [23] B. Yin, J. Wang and W. Wei, Population reproductive characteristics of plateau pika in alpine meadow ecosystem, Acta Theriol. Sin., 24 (2004), 222-228. [24] Y. Zhang, X. Yuan and D. Niu, Response of plateau pika burrow density to vegetation management in an alpine meadow, Maqu County, Gansu, Acta Prataculturae Sin., 25 (2016), 87-94. [25] X. Zhao and X. Zhou, Ecological basis of alpine meadow ecosystem management in Tibet: Haibei alpine meadow ecosystem research station, Ambio, 28 (1999), 642-647. [26] H. Zhou, L. Zhou and X. Zhao, Stability of alpine meadow ecosystem on the Qinghai-Tibetan Plateau, Chin. Sci. Bull., 51 (2006), 320-327. doi: 10.1007/s11434-006-0320-4.
A bifurcation diagram of Model (1) with $r_{1} = 0.227$, $\alpha = 4000$, $\mu = 0.001$, $r_{2} = 0.95$, $K = 4000$, $\lambda = 0.3225$, $q = 0.36$, for $p$ varying from $6$ to $12$. The solid curves represent stable equilibria, while the dotted curves represent unstable equilibria
A bifurcation diagram of Model (1) with $r_{1} = 0.227$, $\alpha = 1000$, $\mu = 0.0001$, $r_{2} = 0.95$, $K = 4000$, $\lambda = 0.3225$, $q = 0.36$ and $p$ varying from $1.2$ to $2.2$. The solid curves represent stable equilibria or limit cycle, while the dotted curves represent unstable equilibria
Two solution trajectories of (1) are converging to a stable limit cycle. Parameter values used ere $r_{1} = 0.227$, $\mu = 0.0001$, $r_{2} = 0.95$, $K = 4000$, $\lambda = 0.3225$, $q = 0.36$, $\alpha = 1000$ and $p = 2$; Two sets of initial conditions are $(3500,1000)$ and $(1500,1000)$
Two stable equilibria $E_{0}$ and $E_{2}$ coexist. Here $r_{1} = 0.227$, $\alpha = 1000$, $p = 1.68$, $\mu = 0.0001$, $r_{2} = 0.95$, $K = 4000$, $\lambda = 0.3225$, $q = 0.36$
The outer closed curve $L$: $OE_{0}FGHO$ around $E_{2}$
The default values of parameters in simulations
 Parameter Value Reference $r_{1}$ $0.227\;year^{-1}$ [8] $K$ $4000\;kg$ [1,8,9,22] $q$ $0.36\;kg\cdot head^{-1}\cdot year^{-1}$ [23] $r_{2}$ $0.95\;year^{-1}$ [19,20] $p$ $1\sim12\;year^{-1}$ [13,14,17,24] $\lambda$ $0.32256\;kg\cdot head^{-1}$ [10]
 Parameter Value Reference $r_{1}$ $0.227\;year^{-1}$ [8] $K$ $4000\;kg$ [1,8,9,22] $q$ $0.36\;kg\cdot head^{-1}\cdot year^{-1}$ [23] $r_{2}$ $0.95\;year^{-1}$ [19,20] $p$ $1\sim12\;year^{-1}$ [13,14,17,24] $\lambda$ $0.32256\;kg\cdot head^{-1}$ [10]
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