STABILITY IN A MODEL FOR GENETICALLY ALTERED MOSQUITOS WITH PERIODIC PARAMETERS

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1 STABILITY IN A MODEL FOR GENETICALLY ALTERED MOSQUITOS WITH PERIODIC PARAMETERS Hubertus F. von Bremen and Robert J. Sacker Department of Mathematics and Statistics, California State Polytechnic University, Pomona, USA, hfvonbremen@csupomona.edu Department of Mathematics, University of Southern California, Los Angeles, USA, rsacker@math.usc.edu Abstract: In this paper we introduce seasonal variability into a discrete-time population dynamics model given by Jia Li for genetically altered and wild mosquitoes. This is done by allowing the birth and survival rates to vary periodically. Using the new model we show that under certain conditions the system will have a globally asymptotically stable solution. Keywords: Periodic difference equation, global stability, genetically altered mosquitoes.. INTRODUCTION Mosquitoes transmit many fatal and debilitating diseases. Malaria is the major killer in sub-saharan Africa with more than one million deaths attributed to it every year. Yellow fever, dengue fever, West Nile virus, encephalitis and filariasis are additional mosquito-borne diseases that have a significant impact on populations worldwide. Recently a significant amount of research has been conducted to genetically modify mosquitoes in order to block the transmission of disease generating parasites. Catteruccia et al [] state that success in genetically modifying the Mediterranean fruit fly, the yellow fever mosquito (Aedes aegypti) has been achieved. Researchers have proposed the release of genetically modified mosquitoes into the wild. The altered mosquitoes would interact with the wild ones and reproduce. This development has fueled the need to create and study mathematical models of the population dynamics of wild and genetically modified mosquitoes. Jia Li [] has recently proposed a promising discrete-time mathematical model for populations consisting of wild and genetically altered mosquitoes. Li [] developed a two species model having a hybrid Ricatti/Ricker type nonlinearity and provided sufficient conditions guaranteeing the existence of locally asymptotically stable fixed point for mosquitoes with a constant mating rate. We [] recently showed that under less restrictive conditions the fixed point is actually globally asymptotically stable with respect to initial populations in which both species are present. We also investigated the case when the mortality rates (fixed as well as density dependent) are different for the two species, and showed that the model becomes very sensitive to small changes in the reproductive parameters. More recently, we showed [] that for proportional mating rates, under a certain set of conditions, the system will have two positive fixed points. The positive fixed point with the largest value is locally asymptotically stable, provided certain conditions are satisfied, and the other positive fixed point is unstable. In the proportional mating rate case the origin is also a fixed point and it is locally asymptotically stable. This creates a "threshold" effect, or allee effect, so that very small initial populations tend to die out. In this paper we present an extension of the population dynamics model (see [], [], []) that takes into account the effect of seasonal variability by allowing the parameters to be periodic. We consider the cases of proportional and constant mating rates where the birth and mortality rates are periodic, but with equal mortality rates for the wild and the genetically altered mosquitoes. The population dynamics model for genetically altered and wild mosquitos with periodic parameters is given in section. In section we consider the ratio dynamics and we show that the under some conditions the ratio dynamics produces a globally asymptotically stable periodic solution. The periodic solutions of the original system are explored in sections and, for the constant mating rate case and the proportional mating rate, respectively.. MODEL FOR THE POPULATION OF MOSQUITOES An extension of the discrete-time mathematical model for populations consisting of wild and genetically altered mosquitoes proposed by Jia Li [] for systems with periodic parameters is considered in this paper. The following description closely follows []. Let x n be the number of wild mosquitoes present at generation n. The population dynamics of the wild mosquitoes is described by the difference equation x n+ = f(n, x n )s(n, x n )x n, () where f is the birth function (per-capita rate of offspring production) and s is the survival probability (fraction of the offspring that survive). The survival probability is assumed to have a Ricker-type form s(n, x n ) = e d(n) k(n)xn. In the model by Jia Li the survival parameters d and k were fixed and did not depend on the generation n. In the formulation here presented, it will be assumed that the parameters are generation dependent and periodic. Let y n be the number of genetically altered mosquitoes present at generation n, and assume that before the wild and altered mosquitoes interact, the dynamics of the altered mosquito population is similar to the one of the wild mosquitoes. Once the altered mosquitoes are released into the wild mosquito habitat, the mosquito populations are governed by the system of difference equations

2 x n+ = f (n, x n, y n )x n e d(n) k(n)(xn+yn), () y n+ = f (n, x n, y n )y n e d(n) k(n)(xn+yn). It is assumed that both wild and altered mosquitoes have the same survival probability e d(n) k(n)(xn+yn). For x n, y n and (x n, y n ) (, ) the birth rate functions f and f are given by f (n, x n, y n ) = c(n n ) α (n)x n + β (n)y n, () f (n, x n, y n ) = c(n n ) α (n)x n + β (n)y n, where c(n n ) is the number of matings per individual, per unit time with N n =. At generation n the number of matings, per individual, with wild mosquitoes is c(n n )x n /( ) and with altered mosquitoes, c(n n )y n /( ). Let α (n) be the number of wild offspring that a wild mosquito produces through mating with a wild mosquito, and β (n) be the number of wild mosquitoes produced through mating with an altered mosquito at generation n. Similarly, α (n) and β (n) are the number of altered mosquitoes produced by the mating of altered mosquitoes with wild and altered mosquitoes respectively at generation n. Combining () and () gives the following set of difference equations that govern the interacting populations of wild and altered mosquitoes x n+ = () c(n n ) α (n)x n + β (n)y n x n e d(n) k(n)(xn+yn), y n+ = c(n n ) α (n)x n + β (n)y n y n e d(n) k(n)(x n+y n ). The mating rate depends on the population density. When the population is relatively small the mating rate will be assumed to be proportional to the total population, N n =, that is, c(n n ) = c N n. Once the population size exceeds a certain level, we expect the number of matings to saturate, and we assume the mating rate is constant, that is,. In the case of small population sizes we assume that the mating rate is proportional to the total population, and the mating rate becomes c(n n ) = c ( ). Using () and a proportional mating rate, we get x n+ = (a (n)x n + b (n)y n )x n e d(n) k(n)(x n+y n ), () y n+ = (a (n)x n + b (n)y n )y n e d(n) k(n)(x n+y n ), where a i (n) = c α i (n) and b i (n) = c β i (n), for i =,. In the constant mating rate case letting c(n n ) = c, and letting a i (n) = cα i (n) and b i (n) = cβ i (n), for i =,, equation () becomes x n+ = a (n)x n + b (n)y n x n e d(n) k(n)(x n+y n ), () y n+ = a (n)x n + b (n)y n y n e d(n) k(n)(x n+y n ). In () and () we assume that x n, y n and (x n, y n ) (, ) for n.. THE RATIO DYNAMICS AND THE STABILITY OF PERIODIC SOLUTIONS In this section we will study the ratio z n = x n /y n and we will show that the under some conditions the ratio dynamics has a globally asymptotically stable periodic solution. Consider the equations in () or (), then the ratio z n = x n /y n for both systems is given by z n+ = a (n)z n + b (n) a (n)z n + b (n) z n. () Due to the decay survival probability term in () and in () the populations can not grow indefinitely ( is not possible). In the following Lemma we show that () has a unique globally asymptotically stable positive periodic solution. Lemma Suppose a(n), b(n), c(n), d(n) >, z, with a(n), b(n), c(n), d(n) being periodic with period k. Consider the difference equation z n+ = F (n, z n ), (8) F (n, z) = a(n)z + b(n) z c(n)z + d(n). If b(n)/d(n) > and c(n)/a(n) >, then (8) has a unique positive periodic s-cycle which is Globally Asymptotically Stable and is such that s divides k. Proof. The result of this Lemma follows directly from a result given by Elaydi and Sacker in []. Let f n (z) = F (n, z). In order to apply the result in [] we need to show that for each n the functions f n satisfy the following properties: )f n : R + R + is continuous, )f n is concave, and ) There exists z and z such that f n (z ) > z and f n (z ) < z. Property ) follows directly from the positivity of a(n), b(n), c(n), d(n), z. Direct computations show that f n(z) < for all z > (see []), showing that Property ) is satisfied. For each n, f n (z) has the unique positive fixed point z = (d(n) b(n))/(a(n) c(n)). Let z > z and z < z, then due to the fact that f n is concave, positive and continuous for z > we have that f n (z ) > z and f n (z ) < z. This shows that Property ) is satisfied. From Lemma() equation () has a unique globally asymptotically stable periodic orbit. Let z n zˆ n, so we have that y n /x n / zˆ n = r n, where r n is a unique globally asymptotically stable periodic orbit.

3 The periodic point r(n) for () gives rise to an invariant set of lines in the (x, y) plane, i.e., the union S = S n of the collection of lines {S n }, where S n = {(x, y) : y/x = r(n)}. From the global asymptotic stability of the periodic orbit r n we then have that y n /x n r(n), i.e. the ω-limit set of any point(x, y) with x >, y > lies in the union of the lines S n. Then to study the solutions in S we can set y n = r(n)x n in the first equation of () (or of (), and in the second equation we can set x n = (/r(n))y n. Using these substitutions we get two uncoupled equations (on S n ), as it will be shown in the next two sections.. CONSTANT MATING RATE CASE In this section we briefly explore the stability of the periodic solutions of systems with a constant mating rate. Using the substitutions y n = r(n)x n in the first equation of (), and x n = (/r(n))y n in the second equation,we get the following two uncoupled equations (on S). x n+ = a (n) + b (n)r(n) x n e d(n) k(n)(+r(n))xn, (9) + r(n) y n+ = a (n) + b (n)r(n) y n e d(n) k(n)(+/r(n))y n. + r(n) Using the transformations x n = u n /(k(n)(+r(n))) and y n = v n /(k(n)( + /r(n))) we can rewrite (9) in terms of u and v as u n+ = u n e pu(n) un, () v n+ = v n e pv(n) vn, Population 8 8 Altered Mosquitoes, y n Wild Mosquitoes, x n 8 9 Figure Population level of wild and genetically altered mosquitoes as a function of generation number Population ratio x n /y n where p u (n) and p v (n) are periodic with period k and given by. p u (n) = d(n) + () log[ k(n + )( + r(n + ))(a (n) + b (n)r(n)) k(n)( + r(n)) ], p v (n) = d(n) + log[ k(n + )r(n)( + r(n + ))(a (n) + b (n)r(n)) k(n)r(n + )( + r(n)) ]. The system in () is just two uncoupled Ricker s equations with periodic parameters. The stability of the solutions of the original coupled system () can be studied by considering the stability of the solutions of (). When periodic solutions of the ratio dynamics are stable, and the periodic solutions of the decoupled system are also stable (), then the periodic solutions of the original system are also stable. The following example illustrates the result presented in section (). Consider the system in () with the following period- parameters a = [,, ], a = [,,.], b = [8., 8., 8.], b = [8.,.,.], k = [.,.,.] and d = [.,.,.]. This set of parameters satisfy the condition in Lemma (), that is, a (n), a (n), b (n), b (n) >, with a (n)/a (n) > and b (n)/b (n) >. The population level of wild and genetically altered mosquitoes as a function of generation number Figure Ratio of population level of wild to genetically altered mosquitos as a function of generation number is shown in Figure. The figure was generated using the system in (), with the above-mentioned parameter values and initial conditions of x =. and y =.. The values of the population levels suggest that the system eventually settles to a periodic solution, with period. The ratio z n = x n /y n of the population levels of wild to genetically altered mosquitos for the above system is plotted on Figure as a function of generation number. As expected from Lemma (), the ratio z n = x n /y n eventually settles to a periodic solution with period. It should be noted that convergence of the ratio z n = x n /y n to a periodic solution is not sufficient to guarantee that x n and y n individually converge to a solution with the same period. Consider the above example with the same parameters, except for k = [.,.,.]. The ratio z n = x n /y n is identical to the previous one. However, x n and y n qualitatively behave significantly different, as shown in Figure. We now have a period- solution, as opposed to the earlier period- solution.

4 8 Altered Mosquitoes, y n. Population 8 Wild Mosquitoes, x n q.. q Figure Population level of wild and genetically altered mosquitoes as a function of generation number Figure Level curves of the positive fixed point of the composite map as a function of q and q Let q u (n) = e p u(n) in (), then the u equation can be written as u n+ = u n q u (n)e un. () Next we consider the special case of k =. Suppose q u (n) is period two, that is, q u (n + ) = q u (n) for all n. Let q = q u () and q = q u () with both q and q being positive. To study the period- solutions of () we can consider z n+ = g(z n ), and () g(z n ) = (g g )(z n ), with g i (z) = q i ze z, and i =,. The fixed points of () are the period- solutions of the u equation in (). Note that the v equation in () can be written in the same form as equation (). In the period- Ricker equation in the form given in (), the origin is a fixed point, and for positive values of q u (n) = q < there are no positive fixed points. In the range < q < e, the period- Ricker equation has unique globally asymptotically stable positive fixed points depending on the value of q. In the period- case the origin is also a fixed point of the composite map (), and for < q < e and < q < e the composite map will have at most one positive fixed point. Figure shows level curves of the positive fixed point of the composite map as a function of q and q. The blank region towards the lower left corner of the plot is the region in the q and q parameter space in which the system has no positive fixed point. Level curves of the derivative of the composite map at the positive fixed point as a function of q and q are shown in Figure. In the parameter range shown < q < e and < q < e the magnitude of the derivative is always less than unity. Thus the fixed point of the composite map is stable, and therefore () will have globally asymptotically stable period two solutions. For a q q Figure Level curves of the derivative of the composite map at the positive fixed point as a function of q and q given value of q and q one can use Figure to estimate the value of the fixed point of the composite map and Figure to estimate the value of the derivative of the composite map at that fixed point. We will present more detailed results on the stability of solutions of the periodic Ricker equation in []. These results give parameter regions guaranteeing stability that are larger than the ones provided by Z. Zhou and X. Zou []. See also [8].. PROPORTIONAL MATING RATE CASE In this section we will focus on the proportional mating rate case (for small population size). As before, using the substitutions y n = r(n)x n in the first equation of (), and x n = (/r(n))y n in the second equation,we get the following two uncoupled equations (on S)

5 x n+ = () (a (n) + b (n)r(n))x ne d(n) k(n)(+r)x n, y n+ = (a (n)(/r(n)) + b (n))yne d(n) k(n)(+/r(n))y n.. The x equation in () is of the form x n+ = α(n)x ne β(n)x n, () with α(n) = (a (n) + b (n)r(n))e d(n) and β(n) = k(n)( + r(n)). Note that the y equation in () has the same general form as (). For the special case of period- parameters (α(n) = α and β(n) = β) it was shown in [] that if α > βe, then () will have two positive fixed points. An unstable fixed point will occur at x < /β, and a stable fixed point will occur at x > /β. The fixed point, x, with x > /β will be locally asymptotically stable if < βx <. Since the origin is also a stable fixed point this gives rise to a threshold effect, i.e., very small initial populations tend to die out. Using the transformations x n = u n /β(n) we can rewrite () in terms of u as p p Figure Level curves of the first positive fixed point of the composite map as a function of p and p.8... u n+ = p u (n)u ne u n, () where p u (n) is periodic with period k and given by p u (n) = α(n)β(n+)/β(n). Again, the y equation in () can also be written in same form as (). The following numerical results deal with the special case of k =. Suppose p u (n) is period two, that is, p u (n + ) = p u (n) for all n. Let p = p u () and p = p u () with both p and p being positive. To study the -periodic solutions of () we can consider z n+ = g(z n ), and () g(z n ) = (g g )(z n ), with g i (z) = p i z e z, and i =,. The fixed points of () are the period- solutions of (). As for the case of period- parameters, the origin is also a stable fixed point of the composite map () for the period- case. For small enough parameter values of p and p, the composite map () has no positive fixed point. Thus in this parameter range, both populations of mosquitoes will become extinct. As the parameter values become sufficiently large, the composite map () will have two positive fixed points. Figure shows level curves of the first positive fixed point of the composite map as a function of p and p. The blank region on the plot corresponds to the set of parameter values of p and p for which the origin is the only fixed point. For a fixed combination of parameter values p and p, Figure can be used to obtain an estimate of the value of the first positive fixed point of the composite map corresponding to the chosen parameters. For a fixed combination of parameter values p and p, Figure can be used to estimate the value p p Figure Level curves of the derivative of the composite map at the first fixed point as a function of p and p..

6 . Wild Mosquitoes, x n. p. Population p. Altered Mosquitoes, y n Figure 8 Level curves of the the second fixed point of the composite map as a function of p and p Figure Population level of wild and genetically altered mosquitoes as a function of generation number p p Figure 9 Level curves of the derivative of the composite map at the second fixed point as a function of p and p of the derivative of the composite map at the first positive fixed point. Figure shows level curves of the derivative of the composite map at the first fixed point of as a function of p and p. The derivative of the composite map is always greater than unity in the range of p and p shown. Thus the first fixed point of the composite map is always unstable in the range of p and p shown. Figure 8 shows level curves of the second positive fixed point of the composite map as a function of p and p. As before, the blank region on the figure corresponds to the set of parameter values of p and p for which the origin is the only fixed point. Figure 9 shows level curves of the derivative of the composite map at the second fixed point of as a function of p and p. The derivative of the composite map is less than unity in the range of p and p shown. Thus the second fixed point of the composite map is stable in the range of p and p shown. The following example illustrates the results presented in this and in section (). Consider the system in () with the following period- parameters a = [.,.], a = [.,.], b = [.,.], b = [.,.], k = [.,.] and d = [.,.]. This set of parameters satisfy the condition in Lemma, that is, a (n), a (n), b (n), b (n) >, with a (n)/a (n) > and b (n)/b (n) >. And therefore we conclude that the ratio dynamics of the system has a unique asymptotically stable periodic solution. The population level of wild and genetically altered mosquitoes as a function of generation number is shown in Figure. The figure was generated using the system in (), with the above-mentioned parameter values and initial conditions of x =. and y =.. The values of the population levels suggest that the system eventually settles to a periodic solution, with period. Note that if the initial conditions are chosen sufficiently small, the system will converge to the zero solution. The ratio z n = x n /y n of the population levels of wild to genetically altered mosquitos is plotted on Figure as a function of generation number. As expected from Lemma, the ratio z n = x n /y n eventually settles to a periodic solution with period.. CONCLUSION An extension of the discrete-time population dynamics model (see [], [], []) for genetically altered and wild mosquitoes that takes into account the effect of seasonal variability is presented in this paper. The seasonal variability is introduced to the mathematical model by allowing the birth and survival rates vary periodically. In this paper we consider proportional and constant mating rates with periodic birth and survival (mortality) rates. The survival rates are allowed to be periodic but they are taken to be equal for the two types of mosquitoes. The stability of the solutions of the new model with periodic parameters is studied by considering the ratio dynamics. We provide conditions that guarantee that the ratio dynamics converges to a globally asymptotically stable periodic orbit. In the con-

7 Population ratio z n =x n /y n generation(n) Figure Ratio of population level of wild to genetically altered mosquitos as a function of generation number Control of Autonomous Decision Support Based Systems. Edited by E P Hofer and E Reithmeier, 8,. [] S. Elaydi and R. J. Sacker, "Global stability of periodic orbits of nonautonomous difference equations", J Differential Eq, 8():8-,. [] R.J. Sacker and H.F. von Bremen, "Stability in the periodic Ricker equation", in preparation. [] Z. Zhou and X. Zou, "Stable Periodic Solutions in a Discrete Periodic Logistic Equation", Applied Mathematical Letters, :-,. [8] R. Kon, "Attenuant cycles of population models with preriodic carrying capacity", Journal of Difference Equations and Applications, (-):-,. stant mating rate case we further show that under some additional conditions the system has globally asymptotically stable periodic orbits. In the proportional mating rate case with periodic parameters the origin is a locally asymptotically stable fixed point. This creates a "threshold" effect, or allee effect, so that very small initial populations tend to die out. In the period- parameter case we show that in a certain parameter region the system has one locally unstable and one locally stable positive periodic orbit. ACKNOWLEDGMENTS H. von Bremen is thankful for the support received from the California State Polytechnic University-Pomona, RSCA (Research, Scholarship, and Creative Activity) grant and the generous support from FUNDUNESP. Robert J. Sacker was supported by the University of Southern California, Letters Arts and Sciences Faculty Development Grant. REFERENCES [] Flaminia Catteruccia, Tony Nolan, Thanasis G. Loukeris, Claudia Blass, Charalambos Savakis, Fotis C. Kafatos, and Andrea Crisanti, "Stable germline transformation of the malaria mosquito Anopheles stephensi", Nature, :99-9, June,. [] J. Li, "Simple mathematical models for mosquito populations with genetically altered mosquitos", Math. Bioscience, 89:9 9,. [] R.J. Sacker and H.F. von Bremen, "Global Asymptotic Stability in the Jia Li Model for Gentically Altered Mosquitoes", Appeared in Difference Equations and Discrete Dynamical Systems, Edited by L J S Allen, B Aubach, S Elaydi and R Sacker, 8,. [] R.J. Sacker and H.F. von Bremen, "Some Stability Results in a Model for Genetically Altered Mosquitoes", Proceedings of the th International Workshop on Dynamics and Control. Appeared in Modeling and