Population Dynamics in a Microfounded Predator-Prey Model. Thomas Christiaans. University of Siegen. Discussion Paper No

Transcription

1 Volkswirtschaftliche Diskussionsbeiträge V W L Population Dynamics in a Microfounded Predator-Prey Model Thomas Christiaans University of Siegen Discussion Paper No ISSN x UNIVERSITÄT SIEGEN FACHBEREICH WIRTSCHAFTSWISSENSCHAFTEN

2 Population Dynamics in a Microfounded Predator-Prey Model Thomas Christiaans Department of Economics, University of Siegen, Siegen, Germany christiaans@vwl.wiwi.uni-siegen.de Abstract. This paper analyzes the dynamics of a two-dimensional microfounded predatorprey model. It is shown that the dynamics closely resemble those of a model commonly used in mathematical biology if parameters of the latter are suitably restricted. The positive equilibrium of the microfounded model is globally asymptotically stable for positive initial values largely irrespective of the parameter values chosen. If a version of Allee s Law is included, however, species extinction becomes possible. Keywords: population dynamics; ratio dependence; species extinction JEL-Classification: Q0 Introduction Mathematical biologists have recently extensively studied an alternative to the classical Lotka-Volterra model and its variations. The classical approach has been losing ground since some of its predictions are not in line with many field observations. The alternative theory relies on the so-called ratio-dependent predator-prey models. Kuang and Beretta 998) have analyzed the global dynamic behavior of the following two-dimensional ratio-dependent type predator-prey model: ṅ = n a ṅ = n n ) bn K n mn f n d n mn, ). ) Here, n and n denote the prey and the predator population, respectively, and a, b, d, f, m, and K are positive parameters whose biological interpretation can be found in Kuang and Beretta 998), e.g. These authors have shown that the system possesses a unique and globally asymptotically stable positive equilibrium that is, an equilibrium where both n and n are positive) for positive initial values if f > d and am b. While f > d is a necessary condition for the existence of such an equilibrium, the condition am b is sufficient for its global stability but not necessary for existence. If am < b, the global dynamic behavior of the model changes substantially. E.g., extinction of both species becomes possible. A further analysis of these equations can be found in Hsu et al. 00), where it is shown that even limit cycles and heteroclinic cycles may exist if am< b. It should be noted that the debate on the appropriate type of models is not yet finished and that the ratio-dependent models do also have their critics, cf. e.g. Abrams and Ginzburg 000) and Deng et al. 003).

3 A common feature of conventional predator-prey models is that they are macro approaches in the sense that populations and their development in time are the basic units of analysis. As Eichner and Pethig 004a) put it, these macro approaches neglect the processes at the micro level of preying and being preyed upon which ultimately generate the growth functions describing the population dynamics. Such a microfoundation of predator-prey models based on economic methodology has been initiated by Hannon 976) and further developed by Tschirhart 000), Pethig and Tschirhart 00), and Eichner and Pethig 004b), e.g. The present paper focuses on the following microfounded two-dimensional system introduced by Eichner and Pethig 004b): ) α ṅ = n A r0 α )n e n e n ) z n α )n e n e α n e ṅ = n A n ) z α n α z n n e n e α )n e ) α γ, 3) ) α γ. 4) For i =,, e i, γ i, ñ i and z i as well as r 0 are positive constants, respectively, while 0<α i <. Cf. Eichner and Pethig 004b) for the ecological or economic interpretation of these parameters. The functions A i are defined by { A i n i ) := min, n } i, i =,. 5) ñ i The inclusion of the functions A i corresponds to the idea that the organism s generation of net offspring is the more hampered, the further n i drops below some critical population level ñ i > 0. According to Allee s Law, species i may be called an endangered species if n i < ñ i cf. Berryman, 003). Regarding the microfoundation, equations 3) and 4) are derived in Eichner and Pethig 004b) from the maximization of a net-offspring function B i x i, z i, n i )= A i n i )x α i i z i z i ) α i γ i, i =,, of the representative organism of species i, where 0<α i < and γ i > 0, subject to the resource constraint e i p i z i p i x i, i =,. x i is organism i s intake of biomass of its prey species i and z i is organism i s loss of own biomass to its predator, species i. For i =, x i = x 0 is the demand of organism for a basic resource whose total supply per period is r 0 > 0. e i > 0 is some exogenous lump-sum income of species i and p i is the market price of biomass of species i. Since species is the top predator its biomass price is set p 0. The equilibrium conditions r 0 = n x 0, n z = n x are used to determine the equilibrium values of p 0, p, x 0, x, and z, which are substituted into the net-offspring functions. Equations 3) and 4) then follow by substitution into the dynamic equations ṅ i = B i n i, i =,.

4 Eichner and Pethig 004b) have analyzed the resulting dynamics of a threedimensional food chain by means of numerical simulations. Among their results is that the positive equilibrium, which they have shown to be unique, is approached for all positive initial values if one sets A i n i ), i =,,3. Thus, extinction in the three-dimensional system seems to be impossible unless Allee s Law is explicitly considered. The purpose of the present paper is to give a complete characterization of the dynamics in case of a two-dimensional system with just two species. As a slight generalization of the two-dimensional case, the parameters α i are replaced by β i 0 and the restrictions 0<α i < by α i > 0. Repeating the derivations of Eichner and Pethig 004b) for the two species case yields the differential equations ṅ = n A r0 n ) n ṅ = n A n ) z β ) α β n e n e z β n e n e α n e n α z n n e β n e n e α n e ) β γ, ) α γ. As z β is just a constant in the two-dimensional setting, it will be assumed that β = 0. This assumption simplifies the analysis without substantially altering the results β > 0 would merely necessitate a modification of some parameter assumptions). In this case, the differential equations can be given a slightly simpler appearance: ṅ = n A r0 n ) n ) α β n e n e z β n e n e α n e ṅ = n A n α z e n ) β n e n e α n e ) β γ, 6) ) α γ. 7) A further simplification follows by setting A i n i ), i =,, for the time being. Letting equations 6) and 7) take the form m := α β β e e, f := α e β, c := z e, ) α ) r0 cn e n e β ṅ = n γ, 8) n n mn ) f cn α ṅ = n γ. 9) n mn While equation 9) is now qualitatively equivalent to ) if α =, equation 8) cannot be transformed into equation ) using the present setting. Eichner and Pethig 004a) have shown, however, that equation ) can be derived from a slightly modified micro model of the ecosystem. The remainder of this paper is devoted to an analysis of equations 6) and 7). An interesting question is whether the dynamic properties of these equations are similar to those of equations ) and ). To answer this question, Section starts with an analysis of the simplified equations 8) and 9). The results will then be 3

5 used to consider the original equations 6) and 7) in Section 3. Section 4 provides a concluding comparison of the microfounded and the conventional model. It will be shown that the dynamics of 8) and 9) if f c > γ /α and 0 < α < closely resemble those of ) and ) if f > d and am b. While according to ) and ) extinction is possible if f > d and am < b, extinction in the microfounded model is impossible under reasonable assumptions unless Allee s Law is included, that is, unless equations 6) and 7) instead of 8) and 9) are used. Analysis of Equations 8) and 9). Equilibria and Local Stability As the right-hand sides of 8) and 9) are not defined at n, n ) = 0,0), it will be assumed that ṅ = ṅ = 0 if n = n = 0 by definition. Calculating the limit for lim n,n ) 0,0) shows that under this assumption both ṅ and ṅ are continuous on R = {n, n ) R : n 0, n 0} if 0<α <, although not differentiable at 0,0), cf. Appendix A. A similar argument applies with respect to equations 6) and 7). All assumptions supposed to hold in the sequel are summarized as follows. They will not be repeated each time a proposition is stated. Assumptions. All parameters appearing in equations 8) and 9) as well as in 6) and 7) are positive. In addition, 0<α <, 0<α, and f c > γ /α. The initial values of n and n are non-negative. If n = n = 0, then ṅ = ṅ = 0. These assumptions serve the following purposes: 0<α < ensures continuity of 8) and 9) on R, f c > γ/α is necessary for the existence of a positive equilibrium, and 0<α will be needed when analyzing equation 7). There are three equilibria. To begin with, set n = 0 and n > 0, implying ṅ = 0 and ṅ =γ n. Thus, there is a first trivial) equilibrium E 0 = 0,0) and there can be no other equilibrium where n = 0. It is obvious that 0,0) is stable along the n -axis. Thus, if there is no prey, the predator will become extinct. Setting n = 0 and n > 0 implies ṅ = 0 and ṅ = r α 0 z β nα γ n. It is straightforward that there is a second equilibrium E at n, n )= z β /α ) r 0,0 γ /α that is stable along the n -axis. Thus, if there is no predator, the prey population reaches a steady state that is directly proportional to the size of the basic resource, r 0. 4

6 A third equilibrium, E, entails positive populations of both species. If n > 0 and n > 0, setting ṅ = 0 yields n = f c γ/α n mγ /α = qn. 0) }{{} =:q Hence, a positive equilibrium cannot exist unless the condition f c > γ /α ) holds, which has already been assumed. Since γ /α should usually be a small number in relation to f c, this assumption is rather natural. Upon subsitution of 0) into ṅ = 0 one gets n = r 0 γ /α ) ce qe ) β /α, mq which together with 0) describes the positive equilibrium E. Observe that in any of the three equilibria thus considered the population of both species is proportional to the size of the basic resource, r 0. As for the stability of the positive steady state, the Jacobian of system 8), 9) evaluated at E can be shown to have the following pattern of signs cf. Appendix B): ṅ / n ṅ / n sgnj)=sgn ṅ / n ṅ / n ) = ) Therefore, TrJ)<0 and J >0, implying that the equilibrium E is locally asymptotically stable by the Routh-Hurwitz criterion. These results are summarized in Proposition There are three equilibria. E 0 = 0,0) is locally asymptotically stable along the n -axis, E = r 0 z β /α /γ /α, 0) is locally asymptotically stable along the n -axis, and ) r 0 ce qe ) β /α E =, q r ) ) 0 ce qe ) β /α mq mq γ /α is locally asymptotically stable. γ /α ). Global Stability The global dynamic behavior of solutions can be determined using the phase diagram, whose derivation is as follows. From 0), the isocline ṅ = 0 is a positively sloped straight line through the origin unionized with the n -axis) in n, n )-space if conditon ) holds. Setting ṅ = 0 and solving for n yields the following expression unionized with the n -axis) for the isocline ṅ = 0 : β e n z γ /β n /r 0 ) α /β n = e α β )γ /β n /r 0 ) α /β β z e. 3) 5

7 Considering the denominator, it is straightforward that denominator 0 n r 0 β α β z ) β /α γ /α =: n. 4) Thus, there is a vertical asymptote at n. n > 0, numerator 0 n r 0 z β /α Similarly, under the precondition that γ /α =: n. 5) Therefore, the isocline cuts the n -axis at the equilibrium E where n = n > n and at the equilibrium E 0, where n = 0). Comparing the sign patterns of the numerator and the denominator shows that the isocline lies below the n -axis for 0<n < n and n > n, while it lies above the axis if n < n < n. n ṅ = 0 ṅ = 0 E E0 n E n Figure. Phase Diagram of Equations 8), 9) As has been shown in Appendix B, the cross partial derivatives of 8) and 9) are ṅ n < 0 and ṅ n > 0, respectively. These partial derivatives give rise to the and signs indicating the directions of motion off the zero isoclines. Putting all this information together yields the phase diagram. For the sake of completeness, it is shown in Appendix C that the slope of the isocline ṅ = 0 is negative if n < n < n. Due to the existence of the vertical asymptote, there is one specialty to be taken care of. Above but near the isocline ṅ = 0 in the region where n is negative, ṅ < 0. As ṅ > 0 below ṅ = 0 in the positive region, the question arises where the sign of ṅ changes. Inspection of equation 8) shows that a sign change off the isocline ṅ = 0 derived from 3) or from n = 0) is possible only if 8) has a vertical asymptote at n mn = 0. Thus, if n > 0 the sign change must occur in the irrelevant region where n < 0. 6

8 The phase diagram reveals that the equilibrium E is globally asymptotically stable for strictly positive initial values. The proof relies on the fact that it is always possible to draw a rectangular closed region from which the trajectories cannot escape cf. the dashed rectangle in Figur ). This proves that the differential equations 8) and 9) have a solution defined for all t 0 Hirsch and Smale, 974, p. 7). According to the Generalized Poincaré-Bendixson Theorem cf. Perko, 996, p. 43), any limit point of trajectories must be an equilibrium if there exists neither a closed orbit nor a separatrix cycle. As the considered region contains just one equilibrium which is locally asymptotically stable, there is no separatrix cycle. Closed orbits are ruled out by the direction of movements in the four regions separated by the zero-isoclines, or, more rigorously, by Dulac s criterion cf. Appendix D). It follows that there is just one possible limit point of trajectories if t, the equilibrium E. This proves global stability. Notice that Figure gives a qualitatively complete picture of the dynamics of system 8), 9). If both initial values are positive, the equilibrium E will be reached and both species will survive. If the intial value of the predator species is zero, a positive inital population of the prey species will reach the equilibrium E. Finally, if the initial value of the prey species is zero, E 0 will be reached and the predator species becomes extinct. The analysis is summarized in Proposition Each trajectory of system 8), 9) converges to an equilibrium. If n 0)=0< n 0), lim t n t ), n t ))=E 0. If n 0)>0= n 0), lim t n t ), n t ))= E. If n 0)>0 and n 0)>0, lim t n t ), n t ))=E. 3 Analysis of Equations 6) and 7) 3. Partitioning the Phase Diagram Using similar parameters as in equations 8) and 9), equations 6) and 7) read ) α ) ṅ = n A r0 cn e n e β n ) γ, 6 ) n n mn ) ṅ = n A f cn α n ) γ, 7 ) n mn where A i n i ) as defined in 5) is repeated here for convenience: { A i n i ) := min, n } i, i =,. ñ i As a first step, observe that the phase diagram can now be partitioned into three regions depending on the values of ñ i. As long as n i ñ i, the phase diagram applies as before. For the vector field generated by 8) and 9) always points inwards any closed rectangular region lying in the positive quadrant, cf. Figure. If n i < ñ i for i = and/or i =, the dynamics will change. The result will depend on the relative position of the equilibrium E. Figure indicates the situation for the case where the equilibrium values of n and n both exceed ñ and ñ, respectively. 7

9 n ṅ = 0 ṅ = 0 E ñ E0 ñ n E n Figure. Partitioning the Phase Diagram It remains to determine the shape of the isoclines in the regions to the left of ñ and below ñ. As there are a lot of feasible parameter configurations giving rise to various details of the implied dynamics, this section solely relies on the analysis of representative phase diagrams without analytically analyzing local stability or instability of equilibria, which is straightforward in most cases, however. Similarly, the exclusion of closed orbits is not explicitely considered as the phase diagrams reveal that trajectories are always trapped in either the basin of attraction of the positive equilibrium or in a region where at least one species eventually becomes extinct. 3. Small Predator Population Below ñ, the shape of the isocline ṅ = 0 is left unchanged, while ṅ = 0 can be solved for n to yield γ ñ ) /α mn n =, 6) f cn /α γ ñ ) /α which is positive at n = ñ due to ). At n a := γ ñ /f c) α < ñ, there is a horizontal asymptote. As n declines further, n in 6) becomes negative. In addition, it is shown in Appendix E that α is a sufficient condition for the slope to be negative. Thus, the relevant part of the isocline has the shape shown in Figure 3. There emerges a new equilibrium E 3, which is unstable, however. As the direction arrows indicate, the equilibrium E may even be approached if n < ñ, that is, if the predator is an endangered species. Extinction of the endangered species is certain, however, if n n a. The system then approaches the equilibrium E on the n -axis, where only the prey survives. Regarding the policy implications of the complete model of Eichner and Pethig 004b), it should be noted that humans may influence the ecosystem by variation of the size of the basic resource, r 0. While the isocline ṅ = 0 is independent of r 0, ṅ = 0 will shift to the left as r 0 decreases. It is therefore interesting to analyze the behavior of the system as E vanishes sliding along ṅ = 0 when r 0 decreases. Such 8

10 n ṅ = 0 ṅ = 0 E ñ E 3 n a E0 E n Figure 3. Dynamics in Case of n < ñ a case, where the equilibrium E does not exist as it would involve an equilibrium value of n which is smaller than ñ, is shown in Figure 4. As the direction arrows indicate, the equilibrium E will be approached. Notice, however, that this result presupposes that n ñ, as the modifications of the dynamic system arising if n < ñ have not been taken into account yet. n ṅ = 0 ṅ = 0 ñ n a E0 E n Figure 4. Dynamics in Case of n < ñ if E Vanishes 3.3 Small Prey Population To the left of ñ, the shape of the isocline ṅ = 0 is left unchanged, while ṅ = 0 from 6 ) can be solved for n to yield n = β e n z n /β γ ñ ) /β n /r 0 ) α /β e α β )γ ñ ) /β n /r 0 ) α /β β z e n /β. 7) Recall that 0<α < and β > 0. As to the denominator, it is straightforward that denominator 0 n r α /α ) β 0 z α β ) β /α ) γ ñ ) /α ) =: na. 8) 9

11 Thus, there is a vertical asymptote at n a. Similarly, numerator 0 n r α /α ) 0 z β /α ) γ ñ ) =: /α ) n0. 9) Thus, the isocline cuts the n -axis at n 0 < na. Similar considerations as in Section. lead to the graph of the isocline shown in Figure 5. As to the and signs beneath the isocline, the remarks given in Footnote apply analogously. n n 0 n a n Figure 5. The Isocline ṅ = 0 if n < ñ With respect to the dynamics, it is important whether a part of the isocline 7) in the positive region lies to the left of ñ. Setting n 0 ñ shows that this condition is equivalent to ñ n : n 0 ñ ñ n. Thus, if ñ < n, there is a part of 7) for n > n 0 lying to the left of ñ. Similarly, setting n a ñ yields n a ñ ñ n. Therefore, the vertical asymptote of 7) lies to the left of ñ if and only if ñ lies to the left of the asymptote at n of equation 3). Proceeding under the reasonable assumption that ñ < n, it follows that two cases must be considered. n ṅ = 0 ṅ = 0 ṅ = 0 ñ n a E0 n 0 n a ñ n n Figure 6. Dynamics in Case of n < ñ if ñ n 0

12 First, assume that ñ n. Figure 6 shows a possible configuration in this case. Other configurations are possible, depending on the relative position of ñ. In case of Figure 6, the predator will become extinct if the initial values of n and n are such that the system starts below the isocline ṅ = 0 or sufficiently far to the left of the ṅ = 0 isocline. Even the prey may become extinct as the equilibrium at n = n 0 is unstable. Second, let n < ñ < n. This implies that the right-hand part and the left-hand part of ṅ = 0 are both valid only up to the point where n = ñ. Moreover, as 6 ) is continuous at n = ñ, the isocline itself is continuous here. One possible configuration is shown in Figure 7, where the positive equilibrium E vanishes. Depending on parameter values, this equilibrium could as well persist. In case of Figure 7, the predator cannot survive in the long run. n ṅ = 0 ṅ = 0 ñ n a E0 ñ n Figure 7. Dynamics in Case of n < ñ if n < ñ < n Putting all information together yields the overall phase diagram, whose appearance depends on the specific values of ñ and ñ. Figure 8 provides one example. n ṅ = 0 ṅ = 0 ṅ = 0 E ñ E 3 E0 E 4 ñ E n Figure 8. A Complete Phase Diagram

13 Although not all possible configurations have been analyzed in detail in this section, it is straightforward to consider the principle possibilities in terms of further phase diagrams. As with the examples given here, it is fairly obvious that the only possible limit points for t are equilibria. Exact proofs could be given along the lines in Section.. The following proposition summarizes the main results. Proposition 3 Let 0,0) < ñ, ñ ) < E. Then each trajectory of system 6) and 7) converges to an equilibrium. The positive equilibrium E is asymptotically stable and its basin of attraction covers at least {n, n ) R n ñ, n ñ }. There exist initial values n 0, ñ ) and n 0, ñ ), respectively, such that the predator or both species eventually become extinct. 4 Discussion If the functions A i n i ) are set A i n i ), the dynamics implied by equations 8) and 9) under conditions f c > γ /α and 0 < α < widely resemble the dynamics of the conventional model ) and ) if f > d and am b, although the differential equations themselves are rather distinct. In both cases, there is a unique and globally stable equilibrium where both species survive if both initial values are positive. If there is no prey, the predator becomes extinct, and if there is no predator, the prey reaches a positive equilibrium. In fact, the phase diagram of ) and ) can be shown to look exactly like Figure if f > d and am > b. Empirically, it would be impossible to distinguish whether a given set of observations was generated by model ) and ) or by 8) and 9). If f > d and am= b, the isocline ṅ = 0 becomes a straight line, leaving the qualitative implications unchanged, however. The condition f c > γ /α is necessary and given other assumptions about parameters following from the micro approach) sufficient for the existence of a positive equilibrium in the microfounded model. In contrast, f > d is just a necessary condition in the conventional model. If am < b, such an equilibrium exists only if f is suitably bounded from above cf. Kuang and Beretta, 998, p. 39). Under such circumstances, equations ) and ) can generate entirely different dynamics. E.g., it is possible that the positive equilibrium E is locally but not globally asymptotically stable, and one or both species could become extinct. Moreover, even limit cycles or heteroclinic cycles are possible cf. Hsu et al., 00). All these cases are excluded in the microfounded model. If f d or f c γ /α, respectively), the positive equilibrium disappears in both models. However, this case is rather irrelevant considering the microfounded model. Notice that f c = α z e /β e ) and that e, e and α, β should reasonably be of comparable magnitude, respectively, while z, the maximum amount of biomass that the prey could use for transactions, should reasonably exceed the natural death rate of the predator, γ <, raised to the power of /α. Thus, the analysis of this case is merely of theoretical interest. The microfounded model thus leaves no room for the empirically relevant case of species extinction, which is possible in case of the conventional model for am< b even if f > d. This result shows that the economics approach to ecology, where species engage in a kind of maximization process, can resemble the dynamics of

14 settled biological models but adds more stability by excluding extinction. This phenomenon can be reintroduced, however, by taking Allee s Law into account. As the analysis of equations 6) and 7) has shown, adding the A i n i )-functions leaves the dynamics unaltered for a region around the equilibrium with positive populations of both species but adds the possibility of extinction if the respective initial values are sufficiently small. Appendix A Continuity of 8) and 9) at the Origin Using the definition of m, it is obvious that n e n e e n mn ) on R, from which Applying some transformations yields ) α r0 cn e n e γ n n n n mn 0 n e n e n mn e. ) β γ ) α r0 n ce ) β γ n. n As n 0, both interval boundaries converge to zero if 0<α <. Thus, ) α ) lim n r0 cn e n e β γ = 0, n,n ) 0,0) n n mn proving continuity of 8) on R if ṅ = 0 for n = n = 0 by definition. Continuity of equation 9) is proven similarly. B Derivation of ) The partial derivative of 8) with respect to n evaluated at ṅ = 0 is ṅ n ṅ =0= ) ) n n me e ) α ) β n mn )n e n e ) α r0 cn e n e β A) n n mn Using the definition of m, it follows that me e = α β )e /β e = α e /β > 0. Substituting into A) shows that the first term in parentheses and therefore the entire expression is negative. The partial derivative of 8) with respect to n, ṅ n = β e ) me ) α ) r0 cn e n e β, n n mn )n e n e ) n n mn is negative since e me < 0. Finally, it is straightforward that the partial derivatives of 9) are and ṅ α mn ) f cn α = > 0 n n n mn ) n mn ṅ n ṅ =0 = α ) mn f cn α < 0, A) n mn n mn proving ). Notice that the signs of the cross partials are determined even off the isoclines. 3

15 C The Slope of 3) Differentiation of n = β e n z γ /β n /r 0 ) α /β e α β )γ /β n /r 0 ) α /β β z e with respect to n, letting D be an abbreviation for the denominator, yields: n n ṅ =0= β e z γ /β n /r 0 ) α /β α e γ /β n /r 0 ) α /β )D D e e α α β )γ /β n /r 0 ) α /β z γ /β n /r 0 ) α /β D. Recall relations 4) and 5). As the denominator is positive if n > n, this expression is negative if β z γ /β n /r 0 ) α /β α γ /β n /r 0 ) α /β )D < e α α β )γ /β n /r 0 ) α /β z γ /β n /r 0 ) α /β. The right-hand side of this inequality is positive as z > γ /β n /r 0 ) α /β if n < n. The lefthand side is negative as n > n implies that D> 0 and β z α β )γ /β n /r 0 ) α /β < 0, respectively. This proves that the isocline ṅ = 0 is negatively sloped if n < n < n. D Exclusion of Closed Orbits Applying Dulac s criterion cf. Perko, 996, p. 6) to equations 8) and 9), there is no closed orbit lying entirely in R = {n, n ) R : n > 0, n > 0} if there exists a function B C R ) such that the trace of the Jacobian of Bṅ,Bṅ ) is not identically zero and does not change sign in R. Now consider the function B= /n n ). The partial derivative A) has been calculated under the assumption that ṅ = 0, which has had just the effect that the term in square brackets in 8) has been omitted in A). Thus, it is straightforward that Bṅ ) = ṅ n n n n ṅ =0 for all n, n ) R. An analogous argument shows that, using A), Bṅ ) = ṅ n n n n ṅ =0 for all n, n ) R. As it follows from Appendix B that both expressions are negative, the trace of the Jacobian of Bṅ,Bṅ ) is negative for all n, n ) R, proving that there are no closed orbits lying entirely in R. E The Slope of 6) Consider the region where n a < n < ñ. As the denominator of the derivative of 6) with respect to n is positive, it suffices to consider the numerator, which is γ ñ ) /α m f cn /α γ ñ ) /α f cn /α mγ ñ ) /α. α 4

16 A sufficient condition for this expression to be negative is that α : α γ ñ ) /α m f cn /α γ ñ ) /α f cn /α mγ ñ ) /α < 0 α )f cn /α α γ ñ ) /α < 0. References Abrams, P. A. and Ginzburg, L. R. 000): The Natur of Predation: Prey Dependent, Ratio Dependent or Neither, Trends in Ecology and Evolution, 5, Berryman, A. A. 003): On Principles, Laws and Theory in Population Ecology, Oikos, 03, Deng, B., Jessi, S., Ledder, G., Rand, A., and Srodulski, S. 003): Biological Control Does Not Imply Paradox A Case Against Ratio-Dependent Models, Discussion Paper, University of Nebraska-Lincoln. Eichner, T. and Pethig, R. 004a): An Analytical Foundation of the Ratio-Dependent Predator-Prey Model, Discussion Paper No. 7-04, University of Siegen. 004b): Economic Land Use, Ecosystem Services and Microfounded Species Dynamics, Discussion Paper No. 6-04, University of Siegen. Hannon, B. 976): Marginal Product Pricing in the Ecosystem, Journal of Theoretical Biology, 56, Hirsch, M. W. and Smale, S. 974): Differential Equations, Dynamical Systems, and Linear Algebra, New York: Academic Press. Hsu, S.-B., Hwang, T.-W., and Kuang, Y. 00): Global Analysis of the Michaelis-Menten- Type Ratio-Dependent Predator-Prey System, Journal of Mathematical Biology, 4, Kuang, Y. and Beretta, E. 998): Global Qualitative Analysis of a Ratio-Dependent Predator- Prey System, Journal of Mathematical Biology, 36, Perko, L. 996): Differential Equations and Dynamical Systems, nd edition, New York: Springer. Pethig, R. and Tschirhart, J. 00): Microfoundations of Population Dynamics, Journal of Bioeconomics, 3, Tschirhart, J. 000): General Equilibrium of an Ecosystem, Journal of Theoretical Biology, 03,

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