Analysis of Prey – Predator System with Prey Population Experiencing Critical Depensation Growth Function

Mohammed Yiha Dawed; Purnachandra Rao Koya; Temesgen Tibebu

doi:doi:10.11648/j.ajam.20150306.23

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Analysis of Prey – Predator System with Prey Population Experiencing Critical Depensation Growth Function

Mohammed Yiha Dawed, Purnachandra Rao Koya, Temesgen Tibebu

Published in American Journal of Applied Mathematics (Volume 3, Issue 6)

Received: 30 November 2015 Accepted: 10 December 2015 Published: 30 December 2015

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Abstract

In this paper we have presented a pair of coupled differential equations to represent a prey – predator system. It is assumed that the growth of the prey population follows critical depensation function and that of the predator population is negative in absence of the prey population. The critical depensation function is special since the growth rate is negative initially but positive later on. This function is stable both at the origin and at the carrying capacity while unstable at the critical mass quantity. The maximum and minimum rates of the critical depensation model are verified. It can be interpreted here that the prey represent fish and the predator represent a kind of birds that mostly feeding on fish to live. We showed the solution of the model is positive and bounded. The mathematical model of the system consisting of 7 parameters is constructed and shown that the non – dimensionalization decreases the number of model parameters to 4. The deterministic behavior of the model around feasible equilibrium points and criteria of the interior positive equilibrium points and their stability are explained. The trivial equilibrium point is always stable while the two axial equilibrium points and the lone interior equilibrium point are either stable or unstable depending on the conditions imposed on the parameters. The criterion for the existence of the limit cycle and the region of existence of interior equilibrium point are identified. Global stability of interior equilibrium points is also studied. For the interior equilibrium point of the model (i) the region of existence is identified (ii) Dulac’s criteria is applied to find the limit cycle and (iii) Lyaponov function is used to analyze the global stability. Simulation study of the model is conducted in support of the analytical analysis. To solidify the analytical results numerical simulations are provided for hypothetical set of parametric values.

Published in	American Journal of Applied Mathematics (Volume 3, Issue 6)
DOI	10.11648/j.ajam.20150306.23
Page(s)	327-334
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2015. Published by Science Publishing Group

Keywords

Predator, Prey, Critical Depensation, Coupled Differential Equations, Equilibrium Point, Global Stability, Limit Cycle, Lyapunove Function

References

[1]	Mohammed Yiha Dawed, Purnachandra Rao Koya and Ayele Taye Goshu. Mathematical Modelling of Population Growth: The Case of Logistic and Von Bertalanffy Models. Open Journal of modeling and Simulation. 2014, Vol. 2, 113–126. http://dx.doi.org/10.4236/ojmsi.2014.24013.
[2]	Purnachandra Rao Koya, Ayele Taye Goshu and Mohammed Yiha Dawed. Modeling Predator Population assuming that the Prey follows Richards Growth Model. European Journal of Academic Essays 1(9): 42-51, 2014 ISSN (online): 2183-1904, ISSN (print): 2183-3818 www.euroessays.org.
[3]	Purnachandra Rao Koya and Ayele Taye Goshu. Solutions of rate-state equation describing biological growths. American Journal of Mathematics and Statistics, Vol. 3 (2013), No. 6, 305-311, http://dx.doi.org/ 10.5923/j.ajms.20130306.02.
[4]	Shonkwiler Herod R. W. Mathematical Biology – A contemporary Approach, 1996.
[5]	Papa Rao A. V., Lakshmi Narayan K. and Shahnaz Bathul. A three species ecological model with a prey, predator and competitor to the prey and optimal harvesting of the prey. Journal of Advanced Research in Dynamics and Control Systems. Vol. 5, issue1, 2013, Pp. 37-49, Online ISSN: 1943-023X.
[6]	Atul Johri, Neetu Trivedi, Anjali Sisodiya, Bijendra Sing and Suman Jain. Study of prey- predator model with diseased prey. Int. J. Contemp. Math. Sciences. Vol. 7, 2012, No. 10, 489 – 498.
[7]	Temesgen Tibebu Mekonen. Bifurcation Analysis on the Dynamics of a Generalist Predator – Prey system. International Journal of Ecosystem. 2012, 2(3), 38 – 43. DOI: 10.5923/j.ije.20120203.02.
[8]	http://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equation (Accessed on November 2015).
[9]	T. K. Kar and H. Matsuda. Sustainable Management of a Fishery with a Strong Allee Effect. Trends in Applied sciences research. 2(4), 271 – 283, 2007. ISSN 1819-3579 C 2007 Academic Journals Inc.
[10]	M. Bandyopadhyay and J. Chatopadhyay. Ratio Dependent predator prey model: effect of environmental fluctuation and stability.
[11]	Linda J. S. Allen. An Introduction to Mathematical Biology.
[12]	Logan J. D. Applied mathematics – A contemporary approach. J. Wiley and Sons, 1987.
[13]	Population dynamics –fisheries economics accessed (on November 2015).

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APA Style

Mohammed Yiha Dawed, Purnachandra Rao Koya, Temesgen Tibebu. (2015). Analysis of Prey – Predator System with Prey Population Experiencing Critical Depensation Growth Function. American Journal of Applied Mathematics, 3(6), 327-334. https://doi.org/10.11648/j.ajam.20150306.23

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ACS Style

Mohammed Yiha Dawed; Purnachandra Rao Koya; Temesgen Tibebu. Analysis of Prey – Predator System with Prey Population Experiencing Critical Depensation Growth Function. Am. J. Appl. Math. 2015, 3(6), 327-334. doi: 10.11648/j.ajam.20150306.23

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AMA Style

Mohammed Yiha Dawed, Purnachandra Rao Koya, Temesgen Tibebu. Analysis of Prey – Predator System with Prey Population Experiencing Critical Depensation Growth Function. Am J Appl Math. 2015;3(6):327-334. doi: 10.11648/j.ajam.20150306.23

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@article{10.11648/j.ajam.20150306.23,
  author = {Mohammed Yiha Dawed and Purnachandra Rao Koya and Temesgen Tibebu},
  title = {Analysis of Prey – Predator System with Prey Population Experiencing Critical Depensation Growth Function},
  journal = {American Journal of Applied Mathematics},
  volume = {3},
  number = {6},
  pages = {327-334},
  doi = {10.11648/j.ajam.20150306.23},
  url = {https://doi.org/10.11648/j.ajam.20150306.23},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20150306.23},
  abstract = {In this paper we have presented a pair of coupled differential equations to represent a prey – predator system. It is assumed that the growth of the prey population follows critical depensation function and that of the predator population is negative in absence of the prey population. The critical depensation function is special since the growth rate is negative initially but positive later on. This function is stable both at the origin and at the carrying capacity while unstable at the critical mass quantity. The maximum and minimum rates of the critical depensation model are verified. It can be interpreted here that the prey represent fish and the predator represent a kind of birds that mostly feeding on fish to live. We showed the solution of the model is positive and bounded. The mathematical model of the system consisting of 7 parameters is constructed and shown that the non – dimensionalization decreases the number of model parameters to 4. The deterministic behavior of the model around feasible equilibrium points and criteria of the interior positive equilibrium points and their stability are explained. The trivial equilibrium point is always stable while the two axial equilibrium points and the lone interior equilibrium point are either stable or unstable depending on the conditions imposed on the parameters. The criterion for the existence of the limit cycle and the region of existence of interior equilibrium point are identified. Global stability of interior equilibrium points is also studied. For the interior equilibrium point of the model (i) the region of existence is identified (ii) Dulac’s criteria is applied to find the limit cycle and (iii) Lyaponov function is used to analyze the global stability. Simulation study of the model is conducted in support of the analytical analysis. To solidify the analytical results numerical simulations are provided for hypothetical set of parametric values.},
 year = {2015}
}

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TY  - JOUR
T1  - Analysis of Prey – Predator System with Prey Population Experiencing Critical Depensation Growth Function
AU  - Mohammed Yiha Dawed
AU  - Purnachandra Rao Koya
AU  - Temesgen Tibebu
Y1  - 2015/12/30
PY  - 2015
N1  - https://doi.org/10.11648/j.ajam.20150306.23
DO  - 10.11648/j.ajam.20150306.23
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 327
EP  - 334
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20150306.23
AB  - In this paper we have presented a pair of coupled differential equations to represent a prey – predator system. It is assumed that the growth of the prey population follows critical depensation function and that of the predator population is negative in absence of the prey population. The critical depensation function is special since the growth rate is negative initially but positive later on. This function is stable both at the origin and at the carrying capacity while unstable at the critical mass quantity. The maximum and minimum rates of the critical depensation model are verified. It can be interpreted here that the prey represent fish and the predator represent a kind of birds that mostly feeding on fish to live. We showed the solution of the model is positive and bounded. The mathematical model of the system consisting of 7 parameters is constructed and shown that the non – dimensionalization decreases the number of model parameters to 4. The deterministic behavior of the model around feasible equilibrium points and criteria of the interior positive equilibrium points and their stability are explained. The trivial equilibrium point is always stable while the two axial equilibrium points and the lone interior equilibrium point are either stable or unstable depending on the conditions imposed on the parameters. The criterion for the existence of the limit cycle and the region of existence of interior equilibrium point are identified. Global stability of interior equilibrium points is also studied. For the interior equilibrium point of the model (i) the region of existence is identified (ii) Dulac’s criteria is applied to find the limit cycle and (iii) Lyaponov function is used to analyze the global stability. Simulation study of the model is conducted in support of the analytical analysis. To solidify the analytical results numerical simulations are provided for hypothetical set of parametric values.
VL  - 3
IS  - 6
ER  -

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Author Information

Mohammed Yiha Dawed

School of Mathematical and Statistical Sciences, Hawassa University, Hawassa, Ethiopia
Purnachandra Rao Koya

School of Mathematical and Statistical Sciences, Hawassa University, Hawassa, Ethiopia
Temesgen Tibebu

School of Mathematical and Statistical Sciences, Debere Birhan University, Debere Birhan, Ethiopia

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APA Style

Mohammed Yiha Dawed, Purnachandra Rao Koya, Temesgen Tibebu. (2015). Analysis of Prey – Predator System with Prey Population Experiencing Critical Depensation Growth Function. American Journal of Applied Mathematics, 3(6), 327-334. https://doi.org/10.11648/j.ajam.20150306.23

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ACS Style

Mohammed Yiha Dawed; Purnachandra Rao Koya; Temesgen Tibebu. Analysis of Prey – Predator System with Prey Population Experiencing Critical Depensation Growth Function. Am. J. Appl. Math. 2015, 3(6), 327-334. doi: 10.11648/j.ajam.20150306.23

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AMA Style

Mohammed Yiha Dawed, Purnachandra Rao Koya, Temesgen Tibebu. Analysis of Prey – Predator System with Prey Population Experiencing Critical Depensation Growth Function. Am J Appl Math. 2015;3(6):327-334. doi: 10.11648/j.ajam.20150306.23

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@article{10.11648/j.ajam.20150306.23,
  author = {Mohammed Yiha Dawed and Purnachandra Rao Koya and Temesgen Tibebu},
  title = {Analysis of Prey – Predator System with Prey Population Experiencing Critical Depensation Growth Function},
  journal = {American Journal of Applied Mathematics},
  volume = {3},
  number = {6},
  pages = {327-334},
  doi = {10.11648/j.ajam.20150306.23},
  url = {https://doi.org/10.11648/j.ajam.20150306.23},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20150306.23},
  abstract = {In this paper we have presented a pair of coupled differential equations to represent a prey – predator system. It is assumed that the growth of the prey population follows critical depensation function and that of the predator population is negative in absence of the prey population. The critical depensation function is special since the growth rate is negative initially but positive later on. This function is stable both at the origin and at the carrying capacity while unstable at the critical mass quantity. The maximum and minimum rates of the critical depensation model are verified. It can be interpreted here that the prey represent fish and the predator represent a kind of birds that mostly feeding on fish to live. We showed the solution of the model is positive and bounded. The mathematical model of the system consisting of 7 parameters is constructed and shown that the non – dimensionalization decreases the number of model parameters to 4. The deterministic behavior of the model around feasible equilibrium points and criteria of the interior positive equilibrium points and their stability are explained. The trivial equilibrium point is always stable while the two axial equilibrium points and the lone interior equilibrium point are either stable or unstable depending on the conditions imposed on the parameters. The criterion for the existence of the limit cycle and the region of existence of interior equilibrium point are identified. Global stability of interior equilibrium points is also studied. For the interior equilibrium point of the model (i) the region of existence is identified (ii) Dulac’s criteria is applied to find the limit cycle and (iii) Lyaponov function is used to analyze the global stability. Simulation study of the model is conducted in support of the analytical analysis. To solidify the analytical results numerical simulations are provided for hypothetical set of parametric values.},
 year = {2015}
}

Copy | Download

TY  - JOUR
T1  - Analysis of Prey – Predator System with Prey Population Experiencing Critical Depensation Growth Function
AU  - Mohammed Yiha Dawed
AU  - Purnachandra Rao Koya
AU  - Temesgen Tibebu
Y1  - 2015/12/30
PY  - 2015
N1  - https://doi.org/10.11648/j.ajam.20150306.23
DO  - 10.11648/j.ajam.20150306.23
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 327
EP  - 334
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20150306.23
AB  - In this paper we have presented a pair of coupled differential equations to represent a prey – predator system. It is assumed that the growth of the prey population follows critical depensation function and that of the predator population is negative in absence of the prey population. The critical depensation function is special since the growth rate is negative initially but positive later on. This function is stable both at the origin and at the carrying capacity while unstable at the critical mass quantity. The maximum and minimum rates of the critical depensation model are verified. It can be interpreted here that the prey represent fish and the predator represent a kind of birds that mostly feeding on fish to live. We showed the solution of the model is positive and bounded. The mathematical model of the system consisting of 7 parameters is constructed and shown that the non – dimensionalization decreases the number of model parameters to 4. The deterministic behavior of the model around feasible equilibrium points and criteria of the interior positive equilibrium points and their stability are explained. The trivial equilibrium point is always stable while the two axial equilibrium points and the lone interior equilibrium point are either stable or unstable depending on the conditions imposed on the parameters. The criterion for the existence of the limit cycle and the region of existence of interior equilibrium point are identified. Global stability of interior equilibrium points is also studied. For the interior equilibrium point of the model (i) the region of existence is identified (ii) Dulac’s criteria is applied to find the limit cycle and (iii) Lyaponov function is used to analyze the global stability. Simulation study of the model is conducted in support of the analytical analysis. To solidify the analytical results numerical simulations are provided for hypothetical set of parametric values.
VL  - 3
IS  - 6
ER  -

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