Skip to main content
SpringerLink
Log in

Lotka-Volterra Type Predator-Prey Models: Comparison of Hidden and Explicit Resources with a Transmissible Disease in the Predator Species

  • Published:
Applications of Mathematics Aims and scope Submit manuscript

Abstract

The paper deals with two mathematical models of predator-prey type where a transmissible disease spreads among the predator species only. The proposed models are analyzed and compared in order to assess the influence of hidden and explicit alternative resource for predator. The analysis shows boundedness as well as local stability and transcritical bifurcations for equilibria of systems. Numerical simulations support our theoretical analysis.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. K. Chakraborty, S. S. Das: Biological conservation of a prey-predator system incorporating constant prey refuge through provision of alternative food to predators: A theoretical study. Acta Biotheoretica 62 (2014), 183–205.

    Article  Google Scholar 

  2. C. W. Clark: Mathematical Bioeconomics: The Optimal Management of Renewable Resources. Wiley-Interscience Publication, John Wiley & Sons, New York, 1990.

    MATH  Google Scholar 

  3. L. M. E. de Assis, M. Banerjee, E. Venturino: Comparing predator-prey models with hidden and explicit resources. Ann. Univ. Ferrara, Sez. VII, Sci. Mat. 64 (2018), 259–283.

    Article  MathSciNet  Google Scholar 

  4. L. M. E. de Assis, M. Banerjee, E.Venturino: Comparison of hidden and explicit resources in ecoepidemic models of predator-prey type. To appear in Comput. Appl. Math.

  5. H. I. Freedman: Deterministic Mathematical Models in Population Ecology. Monographs and Textbooks in Pure and Applied Mathematics 57, Marcel Dekker, New York, 1980.

    Google Scholar 

  6. N. S. Goel, S. C. Maitra, E. W. Montroll: On the Volterra and other nonlinear models of interacting populations. Rev. Mod. Phys. 43 (1971), 231–276.

    Article  MathSciNet  Google Scholar 

  7. M. Haque, B. L. Li, M. S. Rahman, E. Venturino: Effect of a functional response-dependent prey refuge in a predator-prey model. Ecological Complexity 20 (2014), 248–256.

    Article  Google Scholar 

  8. M. Haque, M. S. Rahman, E. Venturino: Comparing functional responses in predatorinfected eco-epidemics models. BioSystems 114 (2013), 98–117.

    Article  Google Scholar 

  9. M. A. Hixon: Species diversity: Prey refuges modify the interactive effects of predation and competition. Theor. Popul. Biol. 39 (1991), 178–200.

    Article  Google Scholar 

  10. A. J. Lotka: Contribution to the theory of periodic reactions. J. Phys. Chem. 14 (1910), 271–274.

    Article  Google Scholar 

  11. A. J. Lotka: Analytical note on certain rhythmic relations in organic systems. Proc. Natl. Acad. Sci. USA 6 (1920), 410–415.

    Article  Google Scholar 

  12. J. D. Murray: Mathematical Biology. Biomathematics 19, Springer, Berlin, 1989. MR

    Google Scholar 

  13. L. Perko: Differential Equations and Dynamical Systems. Texts in Applied Mathematics 7, Springer, New York, 2001.

    Book  Google Scholar 

  14. E. Venturino: The influence of diseases on Lotka-Volterra systems. Rocky Mt. J. Math. 24 (1994), 381–402.

    Article  MathSciNet  Google Scholar 

  15. E. Venturino: Ecoepidemiology: a more comprehensive view of population interactions. Math. Model. Nat. Phenom. 11 (2016), 49–90.

    Article  MathSciNet  Google Scholar 

  16. V. Volterra: Variazioni e fluttuazioni del numero d’individui in specie animali conviventi. Memorie Acdad. d. L. Roma 2 (1927), 31–113. (In Italian.)

    MATH  Google Scholar 

  17. V. Volterra: Variations and fluctuations of the number of individuals in animal species living together. ICES Journal of Marine Science 3 (1928), 3–5.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculdade de Ciências Exatas e Tecnológicas, Universidade do Estado de Mato Grosso, Av. dos Ingas, 3001 Jardim Imperial, Sinop, Brazil

    Luciana Assis

  2. Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, Uttar Pradesh, 208016, India

    Malay Banerjee

  3. Departamento de Ciências Exatas e da Terra, Universidade Federal de Mato Grosso, Cuiabá-Mato Grosso, 78068-600, Brazil

    Moiseis Cecconello

  4. Dipartimento di Matematica “Giuseppe Peano”, Università di Torino, via Carlo Alberto 10, 10123, Torino, Italy

    Ezio Venturino

Authors
  1. Luciana Assis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Malay Banerjee
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Moiseis Cecconello
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ezio Venturino
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Luciana Assis.

Additional information

Ezio Venturino acknowledges the partial support of the program “Metodi numerici nelle scienze applicate” of the Dipartimento di Matematica “Giuseppe Peano” and Luciana Mafalda Elias de Assis acknowledges the support of UNEMAT.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Assis, L., Banerjee, M., Cecconello, M. et al. Lotka-Volterra Type Predator-Prey Models: Comparison of Hidden and Explicit Resources with a Transmissible Disease in the Predator Species. Appl Math 63, 569–600 (2018). https://doi.org/10.21136/AM.2018.0158-18

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.21136/AM.2018.0158-18

Keywords

MSC

Search

Navigation