# Relationship between the Paradox of Enrichment and the Dynamics of Persistence and Extinction in Prey-Predator Systems

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Systems

## 3. Occurrence of the Paradox of Enrichment

#### 3.1. Occurrence of the Paradox of Enrichment with Holling Type I

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

#### 3.2. Occurrence of the Paradox of Enrichment with Holling Type II

**Theorem**

**3.**

**Proof.**

**Corollary**

**1.**

**Theorem**

**4.**

**Theorem**

**5.**

**Proof.**

## 4. Theoretical Approach to Persistence and Extinction

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

**Proof.**

**Theorem**

**8.**

**Proof.**

**Hypothesis 1**

**(H1).**

**Hypothesis 2**

**(H2).**

**Hypothesis 3**

**(H3).**

**Hypothesis 4**

**(H4).**

**Corollary**

**2.**

**Corollary**

**3.**

**Theorem**

**9.**

**Proof.**

**Theorem**

**10.**

**Proof.**

**Theorem**

**11.**

**Proof.**

**Hypothesis 5**

**(H5).**

**Hypothesis 6**

**(H6).**

**Hypothesis 7**

**(H7).**

**Hypothesis 8**

**(H8).**

**Corollary**

**4.**

**Corollary**

**5.**

## 5. Numerical Simulation

## 6. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

- Liu, H.; Cheng, H. Dynamic analysis of a prey–predator model with state-dependent control strategy and square root response function. Adv. Differ. Equ.
**2018**, 2018, 63. [Google Scholar] [CrossRef] [Green Version] - Gurubilli, K.K.; Srinivasu, P.D.N.; Banerjee, M. Global dynamics of a prey-predator model with Allee effect and additional food for the predators. Int. J. Dyn. Control
**2017**, 5, 903–916. [Google Scholar] [CrossRef] - Keong, A.T.; Safuan, H.M.; Jacob, K. Dynamical behaviours of prey-predator fishery model with harvesting affected by toxic substances. Matematika
**2018**, 34, 143–151. [Google Scholar] [CrossRef] - Heurich, M.; Zeis, K.; Küchenhoff, H.; Müller, J.; Belotti, E.; Bufka, L.; Woelfing, B. Selective predation of a stalking predator on ungulate prey. PLoS ONE
**2016**, 11, e0158449. [Google Scholar] [CrossRef] [PubMed] - Gervasi, V.; Nilsen, E.B.; Sand, H.; Panzacchi, M.; Rauset, G.R.; Pedersen, H.C.; Kindberg, J.; Wabakken, P.; Zimmermann, B.; Odden, J.; et al. Predicting the potential demographic impact of predators on their prey: A comparative analysis of two carnivore–ungulate systems in Scandinavia. J. Anim. Ecol.
**2012**, 81, 443–454. [Google Scholar] [CrossRef] [PubMed] - Mougi, A.; Iwasa, Y. Evolution towards oscillation or stability in a predator-prey system. Proc. Biol. Sci.
**2010**, 277, 3163–3171. [Google Scholar] [CrossRef] [PubMed] - Nowak, E.M.; Theimer, T.C.; Schuett, G.W. Functional and numerical responses of predators: Where do vipers fit in the traditional paradigms? Biol. Rev. Camb. Philos. Soc.
**2008**, 83, 601–620. [Google Scholar] [CrossRef] [PubMed] - Griffen, B.D.; Drake, J.M. Effects of habitat quality and size on extinction in experimental populations. Proc. R. Soc. B Biol. Sci.
**2008**, 275, 2251–2256. [Google Scholar] [CrossRef] [Green Version] - Haberman, R. Mathematical Models Mechanical Vibrations, Population Dynamics, and Traffic Flow; Society for Industrial and Applied Mathematics: Philadelphia, PA, USA, 1998. [Google Scholar]
- Murray, J.D. Mathematical Biology; Springer: New York, NY, USA, 2002; Volume 2. [Google Scholar]
- Jensen, C.X.J.; Ginzburg, L.R. Paradoxes or theoretical failures? The jury is still out. Ecol. Model.
**2005**, 188, 3–14. [Google Scholar] [CrossRef] - Rosenzweig, M.L. Paradox of enrichment—Destabilization of exploitation ecosystems in ecological time. Science
**1971**, 171, 385–387. [Google Scholar] [CrossRef] [PubMed] - Walters, C.J.; Krause, E.; Neill, W.E.; Northcote, T.G. Equilibrium-models for seasonal dynamics of plankton biomass in 4 oligotrophic lakes. Can. J. Fish. Aquat. Sci.
**1987**, 44, 1002–1017. [Google Scholar] [CrossRef] - McCauley, E.; Murdoch, W.W. Predator prey dynamics in environments rich and poor in nutrients. Nature
**1990**, 343, 455–457. [Google Scholar] [CrossRef] - Persson, L.; Johansson, L.; Andersson, G.; Diehl, S.; Hamrin, S.F. Density dependent interactions in lake ecosystems—Whole lake perturbation experiments. Oikos
**1993**, 66, 193–208. [Google Scholar] [CrossRef] - Mazumder, A. Patterns of algal biomass in dominant odd-link vs. even-link lake ecosystems. Ecology
**1994**, 75, 1141–1149. [Google Scholar] [CrossRef] - Fussmann, G.F.; Ellner, S.P.; Shertzer, K.W.; Hairston, N.G., Jr. Crossing the hopf bifurcation in a live predator–prey system. Science
**2000**, 290, 1358–1360. [Google Scholar] [CrossRef] [PubMed] - Cottingham, K.L.; Rusak, J.A.; Leavitt, P.R. Increased ecosystem variability and reduced predictability following fertilisation: Evidence from palaeolimnology. Ecol. Lett.
**2000**, 3, 340–348. [Google Scholar] [CrossRef] - Meyer, K.M.; Vos, M.; Mooij, W.M.; Hol, W.H.G.; Termorshuizen, A.J.; van der Putten, W.H. Testing the Paradox of Enrichment along a Land Use Gradient in a Multitrophic Aboveground and Belowground Community. PLoS ONE
**2012**, 7, e49034. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Oksanen, L.; Fretwell, S.D.; Arruda, J.; Niemela, P. Exploitation ecosystems in gradients of primary productivity. Am. Nat.
**1981**, 118, 240–261. [Google Scholar] [CrossRef] - Freedman, I. Deterministic Mathematical Models in Population Ecology; Marcel Dekker, Inc.: New York, NY, USA, 1980. [Google Scholar]
- Smith, H.L. Competitive coexistence in an oscillating chemostat. SIAM J. Appl. Math.
**1981**, 40, 498–522. [Google Scholar] [CrossRef] - Hutson, V.; Vickers, G.A. Criterion for permanent coexistence of species, with an application to a two-prey one-predator system. Math. Biosci.
**1983**, 63, 253–269. [Google Scholar] [CrossRef] - Freedman, H.; Waltman, P. Persistence in models of three interacting predator-prey populations. Math. Biosci.
**1984**, 68, 213–231. [Google Scholar] [CrossRef] - Waltman, P. Coexistence in chemostat-like models. Rocky Mt. J. Math.
**1990**, 20, 777–807. [Google Scholar] [CrossRef] - Ruan, S.; Freedman, H.I. Persistence in three-species food chain models with group defence. Math. Biosci.
**1991**, 107, 111–125. [Google Scholar] [CrossRef] - Kuang, Y.; Beretta, E. Global qualitative analysis of a ratio-dependent predator-prey system. J. Math. Biol.
**1998**, 36, 389–406. [Google Scholar] [CrossRef] - Kuang, Y. Basic properties of mathematical population models. J. Biomath.
**2002**, 17, 129–142. [Google Scholar] - Hsu, S.-B.; Hwang, T.-W.; Kuang, Y. A ratio-dependent food chain model and its applications to biological control. Math. Biosci.
**2003**, 181, 55–83. [Google Scholar] [CrossRef] [Green Version] - Dubey, B.; Upadhyay, R. Persistence and extinction of one-prey and two-predator system. Nonlinear Anal.
**2004**, 9, 307–329. [Google Scholar] - Gakkhar, S.; Singh, B.; Naji, R.K. Dynamical behavior of two predators competing over a single prey. Biosystems
**2007**, 90, 808–817. [Google Scholar] [CrossRef] [PubMed] - Naji, R.K.; Balasim, A.T. Dynamical behavior of a three species food chain model with beddington-deangelis functional response. Chaos Solitons Fractals
**2007**, 32, 1853–1866. [Google Scholar] [CrossRef] - Upadhyay, R.K.; Naji, R.K. Dynamics of a three species food chain model with crowley-martin type functional response. Chaos Solitons Fractals
**2009**, 42, 1337–1346. [Google Scholar] [CrossRef] - Huo, H.F.; Ma, Z.P.; Liu, C.Y. Persistence and stability for a generalized leslie-gower model with stage structure and dispersal. Abstr. Appl. Anal.
**2009**, 2009, 135843. [Google Scholar] [CrossRef] - Kar, T.; Batabyal, A. Persistence and stability of a two prey one predator system. Int. J. Eng. Sci. Technol.
**2010**, 2, 174–190. [Google Scholar] [CrossRef] - Tian, X.; Xu, R. Global dynamics of a predator-prey system with holling type II functional response. Nonlinear Anal. Model. Control
**2011**, 16, 242–253. [Google Scholar] - Smith, H.L.; Thieme, H.R. Dynamical Systems and Population Persistence; Graduate Studies in Mathematics; AMS: Providence, RI, USA, 2011; Volume 118. [Google Scholar]
- Alebraheem, J.; Abu-Hassan, Y. The Effects of Capture Efficiency on the Coexistence of a Predator in a Two Predators-One Prey Model. J. Appl. Sci.
**2011**, 11, 3717–3724. [Google Scholar] [CrossRef] - Alebraheem, J.; Abu-Hassan, Y. Persistence of Predators in a Two Predators-One Prey Model with Non-Periodic Solution. J. Appl. Sci.
**2012**, 6, 943–956. [Google Scholar] - Alebraheem, J.; Abu-Hassan, Y. Efficient Biomass Conversion and its Effect on the Existence of Predators in a Predator-Prey System. Res. J. Appl. Sci.
**2013**, 8, 286–295. [Google Scholar] - Alebraheem, J.; Abu-Hassan, Y. Dynamics of a two predator–one prey system. Comput. Appl. Math.
**2014**, 33, 767–780. [Google Scholar] [CrossRef] - Alebraheem, J. Fluctuations in interactions of prey predator systems. Sci. Int.
**2016**, 28, 2357–2362. [Google Scholar] - Hsu, S.B. On global stability of a predator-prey system. Math. Biosci.
**1978**, 39, 1–10. [Google Scholar] [CrossRef] [Green Version] - Ameixa, O.M.C.C.; Messelink, G.J.; Kindlmann, P. Nonlinearities Lead to Qualitative Differences in Population Dynamics of Predator-Prey Systems. PLoS ONE
**2013**, 8, e62530. [Google Scholar] [CrossRef] [PubMed] - Abu-Hasan, Y.; Alebraheem, J. Functional and Numerical Response in Prey-Predator System. AIP Conf. Proc.
**2015**, 1651, 3. [Google Scholar] [CrossRef] - Alebraheem, J.; Abu-Hassan, Y. Simulation of complex dynamical behaviour in prey predator model. In Proceedings of the 2012 International Conference on Statistics in Science, Business and Engineering, Langkawi, Malaysia, 10–12 September 2012. [Google Scholar]
- Smout, S.; Asseburg, C.; Matthiopoulos, J.; Fernández, C.; Redpath, S.; Thirgood, S.; Harwood, J. The Functional Response of a Generalist Predator. PLoS ONE
**2010**, 5, e10761. [Google Scholar] [CrossRef] [PubMed] [Green Version]

**Figure 1.**Dynamic behavior of system (1) when $k=1$: (

**a**) time series of two trophics x and y; (

**b**) phase space of two trophics.

**Figure 2.**Dynamic behavior of system (1) when $k=4$: (

**a**) time series of two trophics x and y; (

**b**) phase space of two trophics.

**Figure 3.**Dynamic behavior of system (1) when $k=7$: (

**a**) time series of two trophics x and y; (

**b**) phase space of two trophics.

**Figure 4.**Bifurcation diagram for system (1) with Holling type II, using carrying capacity (k) as the bifurcation parameter.

**Figure 5.**Dynamic behavior of system (2) when $k=1$: (

**a**) time series of three trophics x, y and z; (

**b**) phase space of three trophics.

**Figure 6.**Dynamic behavior of system (2) when $k=2$: (

**a**) time series of three trophics x, y and z; (

**b**) phase space of three trophics.

**Figure 7.**Dynamic behavior of system (2) when $k=3$: (

**a**) time series of three trophics x, y and z; (

**b**) phase space of three trophics.

**Figure 8.**Bifurcation diagram for system (2) with Holling type II, using carrying capacity (k) as the bifurcation parameter.

© 2018 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Alebraheem, J.
Relationship between the Paradox of Enrichment and the Dynamics of Persistence and Extinction in Prey-Predator Systems. *Symmetry* **2018**, *10*, 532.
https://doi.org/10.3390/sym10100532

**AMA Style**

Alebraheem J.
Relationship between the Paradox of Enrichment and the Dynamics of Persistence and Extinction in Prey-Predator Systems. *Symmetry*. 2018; 10(10):532.
https://doi.org/10.3390/sym10100532

**Chicago/Turabian Style**

Alebraheem, Jawdat.
2018. "Relationship between the Paradox of Enrichment and the Dynamics of Persistence and Extinction in Prey-Predator Systems" *Symmetry* 10, no. 10: 532.
https://doi.org/10.3390/sym10100532