## Abstract and Applied Analysis

### Birth Rate Effects on an Age-Structured Predator-Prey Model with Cannibalism in the Prey

#### Abstract

We develop a family of predator-prey models with age structure and cannibalism in the prey population. It consists of systems of $m$ ordinary differential equations, where $m$ is a parameter associated with new proposed prey birth rates. We discuss how these new birth rates give the required flexibility to produce differential systems with well-behaved solutions. The main feature required in these models is the coexistence among the involved species, which translates mathematically into stable equilibria and periodic solutions. The search for such characteristics is based on heuristic predation functions that account for cannibalism in the prey.

#### Article information

Source
Abstr. Appl. Anal., Volume 2015, Special Issue (2015), Article ID 241312, 7 pages.

Dates
First available in Project Euclid: 17 August 2015

https://projecteuclid.org/euclid.aaa/1439816311

Digital Object Identifier
doi:10.1155/2015/241312

Mathematical Reviews number (MathSciNet)
MR3384345

Zentralblatt MATH identifier
06929080

#### Citation

Solis, Francisco J.; Ku-Carrillo, Roberto A. Birth Rate Effects on an Age-Structured Predator-Prey Model with Cannibalism in the Prey. Abstr. Appl. Anal. 2015, Special Issue (2015), Article ID 241312, 7 pages. doi:10.1155/2015/241312. https://projecteuclid.org/euclid.aaa/1439816311

#### References

• N. Bairagi and D. Jana, “Age-structured predator-prey model with habitat complexity: oscillations and control,” Dynamical Systems, vol. 27, no. 4, pp. 475–499, 2012.
• E. Brooks-Pollock, T. Cohen, and M. Murray, “The impact of realistic age structure in simple models of tuberculosis transmission,” PLoS ONE, vol. 5, no. 1, Article ID e8479, 2010.
• A. Ducrot, “Travelling wave solutions for a scalar age-structured equation,” Discrete and Continuous Dynamical Systems. Series B. A Journal Bridging Mathematics and Sciences, vol. 7, no. 2, pp. 251–273, 2007.
• W. Feng, M. T. Cowen, and X. Lu, “Coexistence and asymptotic stability in stage-structured predator-prey models,” Mathematical Biosciences and Engineering, vol. 11, no. 4, pp. 823–839, 2014.
• M. E. Gurtin, “A system of equations for age dependent population diffusion,” Journal of Theoretical Biology, vol. 40, no. 2, pp. 389–392, 1973.
• M. E. Gurtin and R. C. MacCamy, “Some simple models for nonlinear age-dependent population dynamics,” Mathematical Biosciences, vol. 43, no. 3-4, pp. 199–211, 1979.
• T. K. Kar and S. Jana, “Stability and bifurcation analysis of a stage structured predator prey model with time delay,” Applied Mathematics and Computation, vol. 219, no. 8, pp. 3779–3792, 2012.
• D. Levine and M. Gurtin, “Models of predator and cannibalism in age-structured populations,” in Differential Equations and Applications in Ecology, pp. 145–159, 1981.
• M. Marvá, A. Moussaouí, R. B. de la Parra, and P. Auger, “A density-dependent model describing age-structured population dynamics using hawk-dove tactics,” Journal of Difference Equations and Applications, vol. 19, no. 6, pp. 1022–1034, 2013.
• V. Pavlová and L. Berec, “Impacts of predation on dynamics of age-structured prey: allee effects and multi-stability,” Theoretical Ecology, vol. 5, no. 4, pp. 533–544, 2012.
• A. McKendrick, Applications of Mathematics to Medical Problems, Springer, 1997.
• H. I. Freedman, J. W. So, and J. H. Wu, “A model for the growth of a population exhibiting stage structure: cannibalism and cooperation,” Journal of Computational and Applied Mathematics, vol. 52, no. 1–3, pp. 177–198, 1994.
• D. S. Levine, “On the stability of a predator-prey system with egg-eating predators,” Mathematical Biosciences, vol. 56, no. 1-2, pp. 27–46, 1981.
• H. P. Benoît, E. McCauley, and J. R. Post, “Testing the demographic consequences of cannibalism in Tribolium confusum,” Ecology, vol. 79, no. 8, pp. 2839–2851, 1998.
• P. W. Flinn and J. F. Campbell, “Effects of flour conditioning on cannibalism of T. Castaneum eggs and pupae,” Environmental Entomology, vol. 41, no. 6, pp. 1501–1504, 2012.
• M. Fernández, “Cannibalism in dungeness crab Cancer magister: effects of predator-prey size ratio, density, and habitat type,” Marine Ecology Progress Series, vol. 182, pp. 221–230, 1999.
• F. J. Solis and R. A. Ku-Carrillo, “Generic predation in age structure predator-prey models,” Applied Mathematics and Computation, vol. 231, pp. 205–213, 2014.
• F. J. Solis, “Self-limitation, fishing and cannibalism,” Applied Mathematics and Computation, vol. 135, no. 1, pp. 39–48, 2003.
• F. J. Solis and R. A. Ku, “Nonlinear juvenile predation population dynamics,” Mathematical and Computer Modelling, vol. 54, no. 7-8, pp. 1687–1692, 2011.
• D. S. Levine, “Models of age-dependent predation and cannibalism via the McKendrick equation,” Computers and Mathematics with Applications, vol. 9, no. 3, pp. 403–414, 1983. \endinput