Predator Population Dynamics Involving Exponential Integral Function When Prey Follows Gompertz Model

Affiliation(s)

School of Mathematical and Statistical Sciences, Hawassa University, Hawassa, Ethiopia.

School of Mathematical and Statistical Sciences, Hawassa University, Hawassa, Ethiopia.

ABSTRACT

The current study investigates the predator-prey problem with assumptions that interaction of predation has a little or no effect on prey population growth and the prey’s grow rate is time dependent. The prey is assumed to follow the Gompertz growth model and the respective predator growth function is constructed by solving ordinary differential equations. The results show that the predator population model is found to be a function of the well known exponential integral function. The solution is also given in Taylor’s series. Simulation study shows that the predator population size eventually converges either to a finite positive limit or zero or diverges to positive infinity. Under certain conditions, the predator population converges to the asymptotic limit of the prey model. More results are included in the paper.

The current study investigates the predator-prey problem with assumptions that interaction of predation has a little or no effect on prey population growth and the prey’s grow rate is time dependent. The prey is assumed to follow the Gompertz growth model and the respective predator growth function is constructed by solving ordinary differential equations. The results show that the predator population model is found to be a function of the well known exponential integral function. The solution is also given in Taylor’s series. Simulation study shows that the predator population size eventually converges either to a finite positive limit or zero or diverges to positive infinity. Under certain conditions, the predator population converges to the asymptotic limit of the prey model. More results are included in the paper.

Cite this paper

Goshu, A. and Koya, P. (2015) Predator Population Dynamics Involving Exponential Integral Function When Prey Follows Gompertz Model.*Open Journal of Modelling and Simulation*, **3**, 70-80. doi: 10.4236/ojmsi.2015.33008.

Goshu, A. and Koya, P. (2015) Predator Population Dynamics Involving Exponential Integral Function When Prey Follows Gompertz Model.

References

[1] Barnes, B. and Fulford, G.R. (2009) Mathematical Modelling with Case Studies: A Differential Equations Approach using Maple and MATLAB. 2nd Edition, Chapman and Hall/CRC, London.

[2] Logan, J.D. (1987) Applied Mathematics—A Contemporary Approach. J. Wiley and Sons, Hoboken.

[3] Chase, J.M., Abrams, P.A., Grover, J.P., Diehl, S., Chesson, P., Holt, R.D., Richards, S.A., Nisbet, R.M. and Case, T.J. (2002) The Interaction between Predation and Competition: A Review and Synthesis. Ecology Letters, 5, 302-315.

http://dx.doi.org/10.1046/j.1461-0248.2002.00315.x

[4] Chamberlain, S.A., Bronstein, J.L. and Rudgers, J.A. (2014) How Context Dependent are Species Interactions? Ecology Letters, 17, 881-890.

http://dx.doi.org/10.1111/ele.12279

[5] Berlow, E.L. (1999) Strong Effects of Weak Interactions in Ecological Communities. Nature, 398, 330-334.

http://dx.doi.org/10.1038/18672

[6] Gurevitch, J.J., Morrison, A. and Hedges, L.V. (2000) The Interaction between Competition and Predation: A Meta-Analysis of Field Experiments. The American Naturalist, 155, 435-453.

http://dx.doi.org/10.1086/303337

[7] Dawed, M.Y., Koya, P.R. and Goshu, A.T. (2014) Mathematical Modelling of Population Growth: The Case of Logistic and Von Bertalanffy Models. Open Journal of Modelling and Simulation, 2, 113-126.

http://dx.doi.org/10.4236/ojmsi.2014.24013

[8] Koya, P.R., Goshu, A.T. and Dawed, M.Y. (2014) Modelling Predator Population Assuming That the Prey Follows Richards Growth Model. European Journal of Academic Essays, 1, 42-51.

http://euroessays.org/archieve/vol-1-issue-9

[9] Koya, P.R. and Goshu, A.T. (2013) Generalized Mathematical Model for Biological Growths. Open Journal of Modelling and Simulation, 1, 42-53.

http://dx.doi.org/10.4236/ojmsi.2013.14008

[10] Koya, P.R. and Goshu, A.T. (2013) Solutions of Rate-State Equation Describing Biological Growths. American Journal of Mathematics and Statistics, 3, 305-311.

[11] Goshu, A.T. and Koya, P.R. (2013) Derivation of Inflection Points of Nonlinear Regression Curves—Implications to Statistics. American Journal of Theoretical and Applied Statistics, 2, 268-272.

[12] Winsor, C.P. (1932) The Gompertz Curve as a Growth Curve. Proceedings of the National Academy of Sciences of the United States of America, 18, 1-8.

http://dx.doi.org/10.1073/pnas.18.1.1

[1] Barnes, B. and Fulford, G.R. (2009) Mathematical Modelling with Case Studies: A Differential Equations Approach using Maple and MATLAB. 2nd Edition, Chapman and Hall/CRC, London.

[2] Logan, J.D. (1987) Applied Mathematics—A Contemporary Approach. J. Wiley and Sons, Hoboken.

[3] Chase, J.M., Abrams, P.A., Grover, J.P., Diehl, S., Chesson, P., Holt, R.D., Richards, S.A., Nisbet, R.M. and Case, T.J. (2002) The Interaction between Predation and Competition: A Review and Synthesis. Ecology Letters, 5, 302-315.

http://dx.doi.org/10.1046/j.1461-0248.2002.00315.x

[4] Chamberlain, S.A., Bronstein, J.L. and Rudgers, J.A. (2014) How Context Dependent are Species Interactions? Ecology Letters, 17, 881-890.

http://dx.doi.org/10.1111/ele.12279

[5] Berlow, E.L. (1999) Strong Effects of Weak Interactions in Ecological Communities. Nature, 398, 330-334.

http://dx.doi.org/10.1038/18672

[6] Gurevitch, J.J., Morrison, A. and Hedges, L.V. (2000) The Interaction between Competition and Predation: A Meta-Analysis of Field Experiments. The American Naturalist, 155, 435-453.

http://dx.doi.org/10.1086/303337

[7] Dawed, M.Y., Koya, P.R. and Goshu, A.T. (2014) Mathematical Modelling of Population Growth: The Case of Logistic and Von Bertalanffy Models. Open Journal of Modelling and Simulation, 2, 113-126.

http://dx.doi.org/10.4236/ojmsi.2014.24013

[8] Koya, P.R., Goshu, A.T. and Dawed, M.Y. (2014) Modelling Predator Population Assuming That the Prey Follows Richards Growth Model. European Journal of Academic Essays, 1, 42-51.

http://euroessays.org/archieve/vol-1-issue-9

[9] Koya, P.R. and Goshu, A.T. (2013) Generalized Mathematical Model for Biological Growths. Open Journal of Modelling and Simulation, 1, 42-53.

http://dx.doi.org/10.4236/ojmsi.2013.14008

[10] Koya, P.R. and Goshu, A.T. (2013) Solutions of Rate-State Equation Describing Biological Growths. American Journal of Mathematics and Statistics, 3, 305-311.

[11] Goshu, A.T. and Koya, P.R. (2013) Derivation of Inflection Points of Nonlinear Regression Curves—Implications to Statistics. American Journal of Theoretical and Applied Statistics, 2, 268-272.

[12] Winsor, C.P. (1932) The Gompertz Curve as a Growth Curve. Proceedings of the National Academy of Sciences of the United States of America, 18, 1-8.

http://dx.doi.org/10.1073/pnas.18.1.1