D Clark

HL Smith

JC Misra

JF Selgrade

MP Hassell

MRS Kulenovic

S Elaydi

Sinan Kapçak

Ünal Ufuktepe

Advances in Difference Equations

English

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Ufuktepe and Kapçak Advances in Diﬀerence Equations 2013, 2013:79http://www.advancesindifferenceequations.com/content/2013/1/79RESEARCH Open AccessStability analysis of a host parasite modelÜnal Ufuktepe* and Sinan Kapçak*Correspondence:unal.ufuktepe@ieu.edu.trDepartment of Mathematics, I˙zmirUniversity of Economics, Balcova,I˙zmir, TurkeyAbstractHost parasite models are similar to host parasitoid models except that the parasitedoes not necessarily kill the host. Leslie/Gower model (Leslie and Gower in Biometrika47(3/4):219-234, 1960) played a historical role in ecology. We consider the stability ofMisra 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).MSC: 39A11; 92D25Keywords: host-parasite; predator-prey; Beddington model; discrete dynamicalsystems1 IntroductionInteraction (predator-prey, host-parasite, host-parasitoid, mutualism, competition) mod-els of two species in nature, sayxt+ = f (xt , yt),yt+ = g(xt , yt),(.)in general describe the change in the populations (i) by assuming a model of growth forboth species in the absence of the other, and (ii) by describing the eﬀect of their interactionon each other. The simplest model isxt+ = axt + bxtyt ,yt+ = cyt + dxtyt .For example, choosing a > , b < ,  < c < , d >  will give us a predator-prey modelwhere the growth or decline is exponential in the absence of the other species in bothcases (xt is the prey population). Choosing a, c >  and b,d >  gives rise to mutualism,where both species get a boost from living together. The mass-action terms xtyt indicaterandom encounters between the species with a signiﬁcant outcome (eating or infectingone another, etc.). One of the earliest realistic models of two-species interaction was de-veloped by Nicholson and Bailey, who applied it to the parasitoid Encarsia formosa andthe host Trialeurodes vaporariorum. A parasitoid is a parasite that can live on its own asan adult, which then lays eggs into a host, eventually causing its death. The general host-© 2013 Ufuktepe and Kapçak; licensee Springer. This is an Open Access article distributed under the terms of the Creative CommonsAttribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproductionin any medium, provided the original work is properly cited.Ufuktepe and Kapçak Advances in Diﬀerence Equations 2013, 2013:79 Page 2 of 7http://www.advancesindifferenceequations.com/content/2013/1/79parasitoid model isHt+ = rHtf (Ht ,Pt),Pt+ = eHt( – f (Ht ,Pt)),(.)where f (Ht ,Pt) is the fraction of host population that is not parasitized at time t, r is thefecundity of the host, and e is that of the parasite, that is, the average number of successfuleggs laid per individual. Note that the parasitoids die in the absence of the host, and for thisreason, 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-actionprinciple and Poisson distribution of the number of encounters, to be f (Ht ,Pt) = e–cPt (thepositive constant c is called the ‘searching eﬃciency’ of the parasitoid).Cooperation between the two populations is reﬂected by the fact that the transitionfunction for each population is an increasing function of the other population size. Thereare several papers on the two-dimensional systems of diﬀerence equations with rationalnonlinearities. Most of these papers are models for either competition, as in [–], or co-operation, as in [–]. Consider the following system of diﬀerence equations with param-eters A,B >  and initial values x, y ≥ .xn+ =Axnyn + yn,yn+ =Bynxn + xn.(.)This system represents the rule by which two discrete, cooperating populations reproducefrom one generation to the next. The phase variables xn and yn denote population sizesduring the nth generation and the sequence or orbit {(xn, yn) : n = , , , . . .} depicts howthe populations evolve over time [].In Hassell’s model [], the growth factor is of the formF(Ht) =R( + aHt)b,where a,b > . This is the population model we will assume for our host species. Hassellet al. [] collected R and b values for about two dozen species from ﬁeld and laboratoryobservations and noted that the majority of these cases were within the stable region.In this paper we study the following host-parasite model, which was studied by Misraand Mitra [], where the growing host is infected with the parasite:Ht+ =RHt( +Ht)be–cPt ,Pt+ =Ht( – e–cPt).(.)Note that such simpliﬁcations, including the convention a =  in the Hassell model, leadto the interpretation of the ‘population’ variable as a ‘suitable multiple of the population’.Ufuktepe and Kapçak Advances in Diﬀerence Equations 2013, 2013:79 Page 3 of 7http://www.advancesindifferenceequations.com/content/2013/1/792 Fixed points of the system (1.4)In order to ﬁnd the ﬁxed points (H*,P*) of the system (.), we set Ht = Ht+ = H* andPt = Pt+ = P*:H* = RH*( +H*)b e–cP* ,P* =H*( – e–cP*).(.)We ﬁrst observe that for H* = , we have the extinction ﬁxed point (, ) for any valuesof parameters. For H* =  and P* = , we obtain the solution (R b – , ). For H* =  andP* = , from the ﬁrst equation in (.), we haveP* = c ln[ R( +H*)b]. (.)Now, we can see that the parameter R is important for the existence of the ﬁxed pointsother than the extinction ﬁxed point (, ). We have the following cases.Case : R≤ .For this case, we have R b –  ≤  and c ln[ R(+H*)b ] < . Hence, there is no exclusion andcoexistence ﬁxed point for R < .Case : R > .By using the ﬁrst equation in (.), assuming that H* = , we havee–cP* = ( +H*)bR . (.)Combining equation (.) and the second equation in (.), we obtainP* =H*( – ( +H*)bR).Now, we can write the ﬁrst equation in (.) asH* = RH*( +H*)b e–cH*(– (+H*)bR )or( +H*)becH*– cRH*(+H*)b = R, (.)an equation in the variable H*. Let us denotez = F(x) = ( + x)becx– cR x(+x)b .When the graph of F intersects the horizontal line z = R, some ﬁxed points are obtained.Note that x = R b –  is a solution of the equation F(x) = R, which corresponds to the ﬁxedpoint (R b – , ) of the system (.). We investigate if there exist some other intersectionpoints.Ufuktepe and Kapçak Advances in Diﬀerence Equations 2013, 2013:79 Page 4 of 7http://www.advancesindifferenceequations.com/content/2013/1/79Figure 1 The intersection point is the point where F′(x) = 0.Figure 2 z = F(x), z = Rwhere b = 2, c = 0.6, R = 1.4.Setting F ′(x) = , we obtain the equationR(b + c + cx)c( + x + bx) = ( + x)b.For x > , the function on the right-hand side has y-intercept  and is monotonicallyincreasing without bound. On the other hand, the function on the left-hand side is mono-tonically decreasing, has y-intercept R( + b/c) > , and converges to R/( + b) as x → ∞.Thus, there is a unique intersection point which means there exists only one critical point(see Figure ).Since F() = , F ′() = c(R–)+bRR > , F(x) →  as x → ∞, the critical point is a localmaximum. See Figure .In Figure , we know that one of the intersection points is the solution x = R b – . Theother intersection point, say H¯ , may or may not be a component of a positive ﬁxed point.In order to guarantee that the P component, say P¯, is also positive, we solve P* >  inequation (.) and ﬁndH* < R b – .Ufuktepe and Kapçak Advances in Diﬀerence Equations 2013, 2013:79 Page 5 of 7http://www.advancesindifferenceequations.com/content/2013/1/79Thus, there exists a positive ﬁxed point only if the H component of it must be less thanthe number R b – , which means among the two intersection points, the one on the rightmust be R b –  and hence F ′(R b – ) < . Solving this inequality, we obtain the conditionfor the existence of the positive ﬁxed point: R > ( + c )b.Thus, we obtain the following result.Theorem . For the system (.), the following statements hold true.a. If R≤ , then the only ﬁxed point is the extinction ﬁxed point (, ).b. If < R≤( + c)b,then there exist two ﬁxed points: the extinction ﬁxed point (, ) and the exclusionﬁxed point (R b – , ).c. IfR >( + c)b,then there exist three ﬁxed points: extinction ﬁxed point (, ), exclusion ﬁxed point(R b – , ), 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 Jacobian matrix of the system (.) isJ =(R( +H – bH)( +H)––be–cP –cRH( +H)–be–cP – e–cP cHe–cP).At (, ), the Jacobian becomesJ =(R  ).The eigenvalues for the ﬁxed point (, ) are λ = R and λ = . Hence, (, ) is asymptot-ically stable if R < . We now consider the exclusion ﬁxed point.Theorem . For the system (.), the exclusion ﬁxed point (R b – , ) is asymptoticallystable ifmax( cc +  ,b – b)< R– b < .Proof The Jacobian matrix evaluated at this point is given byJ =( + b(– + R– b ) –c(– + R b ) c(– + R b ),),where the eigenvalues are λ =  + b(– + R–b ) and λ = c(– + Rb ). Applying the stabilityconditions |λ| <  and |λ| < , we obtain the desired result. Ufuktepe and Kapçak Advances in Diﬀerence Equations 2013, 2013:79 Page 6 of 7http://www.advancesindifferenceequations.com/content/2013/1/79Note that the condition |λ| <  yields R < ( + c )b, for which there does not exist any co-existence ﬁxed point. When the coexistence ﬁxed point appears, the exclusion ﬁxed pointloses stability.We conﬁrmour result by visual representation of the system for some valuesof parameters: Taking b = ., c = ., R = ., for which the condition in Theorem . issatisﬁed, the exclusion ﬁxed point is locally asymptotically stable (see Figure ). However,taking c = . and leaving the other parameters as they are, the coexistence ﬁxed pointappears and the exclusion ﬁxed point loses stability (see Figure ).Figure 3 Phase portrait of the system (1.4) where b = 1.15, c = 2.2, R = 1.5.Figure 4 Phase portrait of the system (1.4) where b = 1.15, c = 3.2, R = 1.5.Ufuktepe and Kapçak Advances in Diﬀerence Equations 2013, 2013:79 Page 7 of 7http://www.advancesindifferenceequations.com/content/2013/1/79Competing interestsThe authors declare that they have no competing interests.Authors’ contributionsUU and SK studied together on all steps of the main problem. SK wrote the Mathematica codes for the phase potrait ofthe system. All authors read and approved the ﬁnal manuscript.AcknowledgementsDedicated to Professor Hari M Srivastava.Received: 15 November 2012 Accepted: 8 March 2013 Published: 28 March 2013References1. Clark, D, Kulenovik, MRS: A coupled system of rational diﬀerence equations. Comput. Math. Appl. 43, 849-867 (2002)2. Hassell, MP, Comins, HN: Discrete time models for two-species competition. Theor. Popul. Biol. 9, 202-221 (1976)3. Kulenovic, MRS, Nurkanovic, M: Global asymptotic behavior of a two dimensional system of diﬀerence equationsmodeling cooperation. J. Diﬀer. Equ. Appl. 9(1), 149-159 (2003)4. Elaydi, S: An Introduction to Diﬀerence Equations. Springer, Berlin (2000)5. Elaydi, S: Discrete Chaos: With Applications in Science and Engineering, 2nd edn. Chapman & Hall/CRC, Boca Raton(2008)6. Selgrade, JF, Ziehe, M: Convergence to equilibrium in a genetic model with diﬀerential viability between the sexes.J. Math. Biol. 25, 477-490 (1987)7. Smith, HL: Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems.Am. Math. Soc., Providence (1995)8. Misra, JC, Mitra, A: Instabilities in single-species and host-parasite systems: period-doubling bifurcations and chaos.Comput. Math. Appl. 52, 525-538 (2006)doi:10.1186/1687-1847-2013-79Cite this article as: Ufuktepe and Kapçak: Stability analysis of a host parasite model. Advances in Diﬀerence Equations2013 2013:79.

Stability analysis of a host parasite model

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Stability analysis of a host parasite model

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