Stability analysis of a host parasite model
© Ufuktepe and Kapçak; licensee Springer. 2013
Received: 15 November 2012
Accepted: 8 March 2013
Published: 28 March 2013
Host parasite models are similar to host parasitoid models except that the parasite does not necessarily kill the host. Leslie/Gower model (Leslie and Gower in Biometrika 47(3/4):219-234, 1960) played a historical role in ecology. We consider the stability of Misra and Mitra’s model (Misra and Mitra in Comput. Math. Appl. 52:525-538, 2006). We study this system analytically and improve the results of Misra and Mitra (Comput. Math. Appl. 52:525-538, 2006).
where is the fraction of host population that is not parasitized at time t, r is the fecundity of the host, and e is that of the parasite, that is, the average number of successful eggs laid per individual. Note that the parasitoids die in the absence of the host, and for this reason, they have been successfully used for eradicating insect pests. In the Nicholson-Bailey model, the function f is determined, under the assumptions of the mass-action principle and Poisson distribution of the number of encounters, to be (the positive constant c is called the ‘searching efficiency’ of the parasitoid).
This system represents the rule by which two discrete, cooperating populations reproduce from one generation to the next. The phase variables and denote population sizes during the n th generation and the sequence or orbit depicts how the populations evolve over time .
where . This is the population model we will assume for our host species. Hassell et al.  collected R and b values for about two dozen species from field and laboratory observations and noted that the majority of these cases were within the stable region.
Note that such simplifications, including the convention in the Hassell model, lead to the interpretation of the ‘population’ variable as a ‘suitable multiple of the population’.
2 Fixed points of the system (1.4)
Now, we can see that the parameter R is important for the existence of the fixed points other than the extinction fixed point . We have the following cases.
Case 1: .
For this case, we have and . Hence, there is no exclusion and coexistence fixed point for .
Case 2: .
When the graph of F intersects the horizontal line , some fixed points are obtained. Note that is a solution of the equation , which corresponds to the fixed point of the system (1.4). We investigate if there exist some other intersection points.
Thus, there exists a positive fixed point only if the H component of it must be less than the number , which means among the two intersection points, the one on the right must be and hence . Solving this inequality, we obtain the condition for the existence of the positive fixed point: .
Thus, we obtain the following result.
If , then the only fixed point is the extinction fixed point .
then there exist three fixed points: extinction fixed point , exclusion fixed point , and a coexistence fixed point.
3 Stability analysis of the system (1.4)
In this section, the stability of the fixed points is examined.
The eigenvalues for the fixed point are and . Hence, is asymptotically stable if . We now consider the exclusion fixed point.
where the eigenvalues are and . Applying the stability conditions and , we obtain the desired result. □
Dedicated to Professor Hari M Srivastava.
- Clark D, Kulenovik MRS: A coupled system of rational difference equations. Comput. Math. Appl. 2002, 43: 849–867. 10.1016/S0898-1221(01)00326-1MATHMathSciNetView ArticleGoogle Scholar
- Hassell MP, Comins HN: Discrete time models for two-species competition. Theor. Popul. Biol. 1976, 9: 202–221. 10.1016/0040-5809(76)90045-9MATHMathSciNetView ArticleGoogle Scholar
- Kulenovic MRS, Nurkanovic M: Global asymptotic behavior of a two dimensional system of difference equations modeling cooperation. J. Differ. Equ. Appl. 2003, 9(1):149–159.MATHMathSciNetView ArticleGoogle Scholar
- Elaydi S: An Introduction to Difference Equations. Springer, Berlin; 2000.Google Scholar
- Elaydi S: Discrete Chaos: With Applications in Science and Engineering. 2nd edition. Chapman & Hall/CRC, Boca Raton; 2008.Google Scholar
- Selgrade JF, Ziehe M: Convergence to equilibrium in a genetic model with differential viability between the sexes. J. Math. Biol. 1987, 25: 477–490. 10.1007/BF00276194MATHMathSciNetView ArticleGoogle Scholar
- Smith HL: Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems. Am. Math. Soc., Providence; 1995.MATHGoogle Scholar
- Misra JC, Mitra A: Instabilities in single-species and host-parasite systems: period-doubling bifurcations and chaos. Comput. Math. Appl. 2006, 52: 525–538. 10.1016/j.camwa.2006.08.026MATHMathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.