Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Storage effect wikipedia , lookup
Introduced species wikipedia , lookup
Unified neutral theory of biodiversity wikipedia , lookup
Molecular ecology wikipedia , lookup
Biodiversity action plan wikipedia , lookup
Latitudinal gradients in species diversity wikipedia , lookup
Occupancy–abundance relationship wikipedia , lookup
Island restoration wikipedia , lookup
Mathematical Biosciences 194 (2005) 21–36 www.elsevier.com/locate/mbs Traveling wave solutions of a reaction diffusion model for competing pioneer and climax species S. Brown a,* , J. Dockery b, M. Pernarowski b a b Mathematics Department, Humboldt State University, Arcata, CA 95521, United States Department of Mathematical Sciences, Montana State University, Bozeman, MT 59717, United States Abstract Presented is a reaction-diffusion model for the interaction of pioneer and climax species. For certain parameters the system exhibits bistability and traveling wave solutions. Specifically, we show that when the climax species diffuses at a slow rate there are traveling wave solutions which correspond to extinction waves of either the pioneer or climax species. A leading order analysis is used in the one-dimensional spatial case to estimate the wave speed sign that determines which species becomes extinct. Results of these analyses are then compared to numerical simulations of wave front propagation for the model on one and twodimensional spatial domains. A simple mechanism for harvesting is also introduced. Ó 2005 Elsevier Inc. All rights reserved. Keywords: Pioneer-climax model; Reaction-Diffusion; Traveling Waves 1. Introduction In an ecosystem, the competition among plant or animal species for natural resources is important in determining the evolution of the system. For example, each tree in a forest competes with its neighbors for light, space, carbon dioxide, and soil nutrients. Although such competition may * Corresponding author. Tel.: +1 707 826 4248; fax: +1 707 826 3140. E-mail addresses: [email protected] (S. Brown), [email protected] (J. Dockery), pernarow@ math.montana.edu (M. Pernarowski). 0025-5564/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.mbs.2004.10.001 22 S. Brown et al. / Mathematical Biosciences 194 (2005) 21–36 or may not be affected by the species of the neighboring tree, it is almost always affected by the density of the neighboring trees. Similarly, the types of food consumed by competing animals may not be as relevant to their success as the amount of food consumed by their competitor. Such observations have led to the development of population models where speciesÕ per capita growth rate (i.e., fitness) is a function of a weighted average of the populations of all competing species. In the absence of competition, population densities u(t) of individual species are often modeled via the equation du ¼ uf ðuÞ; ð1Þ dt where t is time and f(u) is a fitness function. The sign and monotonicity properties of the fitness function determine the species density-dependent growth characteristics. At large densities, crowding tends to cause a decline in the population so the fitness function f(u) < 0 at larger u. Some species thrive best at low densities. For example, certain varieties of pine and poplar have a fitness which decreases monotonically with the total tree density of the forest. Species whose fitness decreases with population density and have a sole equilibrium are often referred to as ÔpioneerÕ species. To study pioneering fish populations, Ricker [1] used the fitness function: f ðuÞ ¼ erð1uÞ a; Hassell and Comins [2] used the alternate form r f ðuÞ ¼ a ð1 þ buÞp ð2Þ ð3Þ in their two-species competition model. Regardless of the specific functional form, pioneer fitness functions satisfy: f 0 ðuÞ < 0; f ðz0 Þ ¼ 0 ð4Þ for some z0 > 0 (cf. Fig. 1(a)). As their name suggests, pioneer species tend to be good colonizers since their per capita growth rate is maximal at low densities. In contrast, other species have survival and reproduction rates which benefit from increased population densities. Group defense for prey, increased gene pools, enhanced soil nutrients, and photosynthetic adaptation to shade are a few examples of factors which can lead to an increased fitness at higher densities. A species whose fitness g(v) is maximal at intermediate densities Fig. 1. It shows typical fitness functions for (a) pioneer species and (b) climax species. S. Brown et al. / Mathematical Biosciences 194 (2005) 21–36 23 while negative at low and high densities is commonly referred to as a ÔclimaxÕ species. In an agestructure study, Cushing [3] used the climax fitness function: gðvÞ ¼ verð1vÞ a: ð5Þ Selgrade and Namkoong [4–6] used a similar fitness function in a forestry model. Mathematically, climax fitness functions g(v) have two roots w1 and w2, and satisfy (cf. Fig. 1(b)): gðw1 Þ ¼ gðw2 Þ ¼ 0; g0 ðw1 Þ > 0; 0 < w1 < w2 g0 ðw2 Þ < 0: ð6Þ ð7Þ Climax species often thrive in the colonizing presence of pioneering species. For instance, climax species like oak and maple trees benefit from the presence of other tree species through the protection and improved soil conditions they provide. In such ecosystems, the survivability of both the climax and pioneer species depends on the total (rather than individual) species population. Selgrade and Namkoong [4–6] examined the dynamics of such a pioneer-climax ecosystem where the model equations are assumed to have the form: du ¼ uf ðy 1 Þ dt ð8Þ dv ð9Þ ¼ vgðy 2 Þ: dt In the pioneer-climax model (8) and (9), u and v denote the pioneer and climax densities and the fitness functions f and g are assumed to satisfy the conditions (4) and (6) and (7), respectively. The variables yi, i = 1,2, are weighted population densities given by: y 1 ¼ c11 u þ c12 v; ð10Þ y 2 ¼ c21 u þ c22 v; ð11Þ where cij > 0 are interaction coefficients which weight the relative importance of total population on each individual speciesÕ fitness. Diagonal and off diagonal elements of the interaction matrix C = [cij] weigh the relative importance of intra-species and inter-species interactions, respectively. In this manuscript, we examine some of the effects of dispersion on competing pioneer and climax species. Letting u(x, t) and v(x, t) be the respective densities at a spatial position x and time t, dispersion can be modeled by including diffusion terms into (8),(9) resulting in the system ou ¼ uf ðy 1 Þ þ D1 r2 u; ot ð12Þ ov ¼ vgðy 2 Þ þ D2 r2 v: ð13Þ ot Here, the diffusion coefficients Di,i = 1,2, reflect a net species dispersion.Throughout, we assume the interaction matrix C has the form: 24 S. Brown et al. / Mathematical Biosciences 194 (2005) 21–36 C¼ c11 1 1 c22 : ð14Þ This assumption is equivalent to the transformations y1 ! y1/c12 and y2 ! y2/c21 (in the case c1250, c2150) and therefore only represents a re-scaling of yi which can be absorbed into the definitions of f and g. This rescaling of the interaction matrix C is commonly used in the pioneer-climax models, see [4–9]. We begin our study in Section 2 by summarizing necessary stability information about the equilibria of the reaction system (8) and (9). In the following Section 3, we use singular perturbation techniques to construct a leading approximation to a traveling wave solution of the associated reaction diffusion system (12) and (13) when D2 is small relative to D1. There we also indicate how certain monotonicity properties associated with such solutions can affect the sign of the wave speed. Owen and Lewis [10] look at similar behavior for a prey invasive wave in a predator-prey reaction-diffusion model. As the waves connect equilibria in which one component is zero, this sign determines which species becomes extinct and which survives. The stability of such waves and the reversal of the sign of the wave speed is demonstrated numerically in Section 4. In this same section, we also show how the one dimensional analysis can be used to predict the sign of plane wave speeds for propagating fronts on two dimensional domains. Lastly, in the discussion, we numerically illustrate how for other parameter sets, the same model under consideration can exhibit a vast variety of qualitatively different spatio-temporal patterns. 2. Dynamics of the spatial homogeneous model In this section we summarize results relating to the location and stability of equilibria of the spatially homogeneous model in (8) and (9). A complete categorization of the stability of the equilibria for this case can be found in Buchanan [7]. From (8) and (9) it is evident that u = 0 and v = 0 are trivial nullclines for the pioneer and climax species, respectively. The non-trivial nullcline for the pioneer species is given by the linear equation c11 u þ v ¼ z0 ; while those for the climax species are given by the two linear equations u þ c22 v ¼ wi ; i ¼ 1; 2: At the onset, we note and subsequently consider only biologically significant equilibria whose components are both non-negative. Since the nullclines are linear, the possible nullcline configurations in the positive quadrant depend only on the order in which the nullclines intercept each coordinate axis. Since we assume 0 < w1 < w2, there are only three cases to consider for the vintercepts: 0 < z0 < w1/c22 < w2/c22, 0 < w1/c22 < z0 < w2/c22, and 0 < w1/c22 < w2/c22 < z0. For the uintercepts, there are three analogous parameter categorizations. Thus, given the possible u and v intercept arrangements, there is a S. Brown et al. / Mathematical Biosciences 194 (2005) 21–36 25 total of nine possible non-degenerate nullcline configurations. Buchanan [7] shows that seven of these nine configurations have hyperbolic equilibria whereas in the other two cases Hopf bifurcations are possible. Throughout this paper we will only consider the case 0 < z0 < w1 =c22 < w2 =c22 Fig. 2. Phase planes for the spatially homogeneous model. Dashed and solid lines are u (pioneer) and v (climax) nullclines, respectively. Open circles indicate unstable equilibria whereas dark circles indicate stable equilibria. 26 S. Brown et al. / Mathematical Biosciences 194 (2005) 21–36 and for simplicity assume c22 = 1. Under these more restrictive assumptions, there are only three qualitatively different nullcline configurations. The algebraic constraints for these three cases are indicated in Fig. 2 along with their respective u intercepts arrangements. In Case 1 there are four equilibria, two of which are stable: (0, w2) and (z0/c11,0). In Case 2, there are five rest states, one of which is in the interior of the positive quadrant: ðui ; vi Þ ¼ ððw1 z0 Þ=ð1 c11 Þ; ðz0 c11 w1 Þ=ð1 c11 ÞÞ: This rest state can undergo a Hopf bifurcation as one varies c11, giving rise to periodic orbits, see for example [5,11]. In Case 3 there are six rest states, two of which are in the positive quadrant. The stability of these states depend on the parameters in a more complicated way but in Buchanan [7] it is shown that in both Case 1 and Case 3 the rest states (0, w2) and (z0/c11,0) are always stable. 3. Traveling waves While there has been several papers that deal with pioneer-climax species very little attention has been paid to the interaction of the pioneer-climax populations with the spatial environment. If we are to consider the next step in modeling the pioneer-climax populations interaction then space is a natural way to go. This move toward including the spatial environment in population models is becoming increasingly prominent in ecological studies. For an overview of the challenges in spatial ecology and the techniques of including this aspect in models see [12,13]. In this paper we will model the interaction of a pioneer species with a climax species and include spatial variability in the simplest manner by allowing the species to disperse in the environment via diffusion. We assume that the rate of dispersion for the climax species is much slower than the pioneer species. The model equations considered are, ut vt ¼ uxx þ uf ðc11 u þ vÞ; ¼ 2 vxx þ vgðu þ vÞ; x 2 ð1; þ1Þ; ð15Þ where 0 < 1 is a small parameter and the functions f and g are as described in the previous section. As before, u and v are the population densities of the pioneer and climax, respectively. In this section, we seek a traveling wave solution of (15) with a slow wave speed c which connects the two equilibria ðcz110 ; 0Þ and (0, w2). Since the population of one species is essentially zero in the wake of such solutions they may be considered extinction waves where there is a transition of dominance of one species to the other. The sign of the wave speed is especially important since it determines which species becomes extinct. For the remainder of this manuscript we restrict our attention to Case 1 due to the stability properties of the traveling wave we are seeking. In this case (see Fig. 2) both equilibria ðcz110 ; 0Þ and (0, w2) are stable. Given z0/c11 < w1 in Case 1, we must restrict the value of c11 so that c11 > c11 z0 =w1 . We remark that although subsequent techniques can be modified to prove the existence of a similar traveling wave solutions in Case 2 and Case 3, such waves are unstable for the reaction diffusion system (15). S. Brown et al. / Mathematical Biosciences 194 (2005) 21–36 27 Given these assumptions we let uðx; tÞ ¼ UðzÞ; vðx; tÞ ¼ V ðzÞ; z ¼ x ct in (15) to obtain the following system of second-order ordinary differential equations U 00 þ cU 0 þ Uf ðc11 U þ V Þ ¼ 0; 2 V 00 þ cV 0 þ VgðU þ V Þ ¼ 0: ð16Þ This resulting problem is notably similar to the traveling wave problem considered by [14] in their study of a predator-prey system. Here we use singular perturbation methods to construct a solution of (16) with the asymptotic behaviors lim ðUðzÞ; V ðzÞÞ ¼ ðz0 =c11 ; 0Þ; z!1 ð17Þ lim ðU ðzÞ; V ðzÞÞ ¼ ð0; w2 Þ: z!1 First we consider the singular limit problem by setting = 0 in (16) to obtain U 00 þ Uf ðc11 U þ V Þ ¼ 0; ð18Þ VgðU þ V Þ ¼ 0: Given w2 is a root of g, the second equation in (18) has two solutions: V ¼ h ðU Þ; ð19Þ where h ðU Þ ¼ w2 U; ð20Þ hþ ðU Þ 0: ð21Þ Thus, in the outer regions we must solve U 00 þ Uf ðc11 U þ h ðU ÞÞ ¼ 0 for z<0 U 00 þ Uf ðc11 U þ hþ ðU ÞÞ ¼ 0 for z P 0 with with U ð1Þ ¼ 0; Uð1Þ ¼ z0 =c11 : ð22Þ ð23Þ It is easy to check that under our assumption that c11 > c11 , f(c11U + h(U)) < 0 on the interval (0, z0/c11). If follows that the (U, U 0 )-phase plane for (22) is as shown in Fig. 3(a). Likewise, it Fig. 3. Depicts the phase planes for the outer systems (22) and (23) in (a) phase plane and (b) + phase plane, respectively when c11 > c11 . 28 S. Brown et al. / Mathematical Biosciences 194 (2005) 21–36 is easy to check that for 0 < U < z0/c11, f(c11U + h+(U)) > 0 from which it follows that the phase plane for (23) is as depict in Fig. 3(b). By integrating (22) and (23), one can show that the unstable manifold, W, of (0,0) in the phase plane intersects the stable manifold W+ in the + phase plane at the point (U, U 0 ) = (U , P ) where U is a root of the function Z z0 =c11 Z u sf ðc11 s þ w2 sÞds þ sf ðc11 sÞds ð24Þ F ðuÞ ¼ 0 u 0 and P = U (0) is given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z U P ¼ 2 sf ðc11 s þ w2 sÞds: 0 This intersection is depicted in Fig. 4. Noting, F(0) > 0, F(z0/c11) < 0 and F 0 (U) < 0 by the assumed monotonicity of f in (4), there is a unique root of F, U 2 (0, z0/c11). One can also check that U is a C1 function of c11 which decreases in c11. It follows that there is a unique solution to (22),(23) for which U(z) 2 (0, z0/c11). In summary we have the following result. Proposition 3.1. Under the assumption that f 0 < 0 and that c11 > c11 , there is a unique monotone C1 solution to the outer problem (22) and (23) denoted by UOuter(z). Furthermore, UOuter(0) U , is a decreasing function of c11 with U (c11) ! 0 as c11 ! 1. While the U-component in the outer region is C1, the V component has a jump discontinuity at z = 0, the value of z for which U(z) = U . To smooth out V we introduce a transition (inner) layer about z = 0. In this layer we introduce the stretched variable, n = z/. Letting v(n) = V(z) and u(n) = U(z) in (16) we obtain the layer equations: u00 þ 2 ðcu0 þ uf ðc11 u þ vÞ ¼ 0; v00 þ cv00 þ vgðu þ vÞ ¼ 0: Fig. 4. Intersection of the stable and unstable manifolds of the outer systems. ð25Þ S. Brown et al. / Mathematical Biosciences 194 (2005) 21–36 29 Setting = 0 in (25) we obtain to leading order in : v00 þ cv0 þ vgðU þ vÞ ¼ 0: ð26Þ In the inner layer, v provides a transition from V = h(U ) to V = h+(U ). Thus we seek a solution to (26) satisfying the matching conditions lim vðnÞ ¼ h ðU Þ ¼ w2 U ; ð27Þ lim vðnÞ ¼ hþ ðU Þ ¼ 0: ð28Þ n!1 n!1 For U 2(0, w1), vg(U + v) is qualitatively cubic and thus (26) is the traveling wave equation for the bistable reaction-diffusion equation. For such systems, it is well known [15] that there is a unique value of the wave speed c, say c = c , for which there is a unique (modulo translations) monotone solution of (26) satisfying (27) and (28). The sign of the wave speed is given by Z h ðU Þ vgðU þ vÞdv : ð29Þ signðc Þ ¼ sign 0 We let vinner(n) denote this solution to (26)–(28). An O(1) composite solution to the original system (16) is then obtained by combining the leading order inner and outer solutions. This leads to the composite solution given by U c ðzÞ ¼ U Outer ðzÞ; ( ðU U c ðzÞÞ þ V Inner z V c ðzÞ ¼ V Inner z for z > 0: ð30Þ for z < 0; ð31Þ In conclusion, the composite solution defined in (30),(31) is an O(1) approximate solution to (16) and (17) which is C1 in U and C0 in V. The existence of such an approximate solution does not guarantee the existence of a true solution. However, one can show using geometric singular perturbation techniques discussed in [16,17] that there is a true solution to the above problem near our approximate solution. To do this, one first writes (25) as a first order system, u0 ¼ p; p0 ¼ 2 cp uf ðc11 u þ vÞ; v0 ¼ q; q0 ¼ cq vgðc11 u þ vÞ: ð32Þ Then, the basic idea is to use invariant manifold theory as described by Fenichel [18] to show that there is a transverse intersection of the center-unstable manifold of the rest state (u, u 0 , v, v 0 ) = (0, 0, w2, 0) with the center-stable manifold of the rest state (u, u 0 , v, v 0 ) = (z0/c11, 0, 0, 0) when = 0 and then to show that this intersection persists for > 0 but small. To obtain a transverse intersection one must append the trivial equation c 0 = 0 to the system (32). Such an inclusion also helps to show that the solution as well as the wave speed are locally unique for > 0 and small. Thus, the limiting value of the wave speed as ! 0 is given by c previously defined. A detailed proof can be found in [19]. 30 S. Brown et al. / Mathematical Biosciences 194 (2005) 21–36 The next natural question that arises after we know there is a traveling wave is to ascertain its stability in the reaction-diffusion system (15). This is important since the waves should be stable if one is to observe them experimentally or in numerical simulations. As previously mentioned, our system is very similar to the predatory-prey system studied in [14]. Though we do not explicitly apply their methods here or supply a proof, these similarities may be exploited to show that the traveling wave whose leading order approximate is given by (30) and (31) is indeed stable for the reaction diffusion system (15) when c11 > c11 . In the next section, our numerical simulations at least provide cursory evidence that such waves are stable. Lastly, the above analysis can be used to determine important qualitative information about the leading order approximation to the wave speed c. Recall that this speed is determined uniquely by the heteroclinic solution to the layer equation (26). Given (29) and Proposition 3.1, this solution depends on the location of the layer, U , and in turn on the model parameter c11. We remark that if Z h ð0Þ vgðU þ vÞdv < 0 0 then by decreasing c11 it may be possible to change the sign of the wave speed. Indeed, depending on the details of the pioneer fitness function f, it is possible that Z h ð0Þ vgðU ðc11 þ vÞÞdv > 0: 0 It would then follow from the monotonicity of U mentioned in Proposition 3.1 that there is a value of c11 > c11 , say c011 , such that 8 0 > < > 0 if c11 < c11 ; c ¼ 0 if c11 ¼ c011 ; > : < 0 if c11 > c011 : Thus one could change the direction of invasion simply by changing c11 which represents the intra-pioneer interaction. In particular, if c11 is large enough then it is possible for the pioneer species to invade whereas for smaller values of c11 the climax species would invade. Another way one could change the sign of the wave speed is through harvesting. For instance, suppose that the climax species is harvested at a rate proportional to its population size: vt ¼ 2 vxx þ vðgðu þ vÞ bÞ: One can show U is an increasing function of b. Then if at b = 0 the wave speed is positive it may be possible to change its sign by increasing b. While we are unable to show that this necessarily must happen in general, we will illustrate this possibility in the next section with a specific example. Although this result is mathematically interesting as of yet we have no indication that this change of wave speed occurs naturally. However, it can be used as an indication of a possible management tool. This idea of changing the wave speed and direction to help manage a population is discussed in the article by Owen and Lewis [10]. In this article they found that if the prey S. Brown et al. / Mathematical Biosciences 194 (2005) 21–36 31 equation had a strong Allee growth function with negative growth at low densities, then the addition of predation could cause the net prey growth to become negative, thus reversing the prey invasive wave. 4. Numerical simulations In this section we illustrate how one can use the leading order results of the previous section to predict behaviors in the model. Such predictions will be confirmed by presenting some numerical simulations. In this section, we take the pioneer fitness function to be linear: f ðuÞ ¼ z0 u; ð33Þ and the climax fitness function to be quadratic: gðvÞ ¼ ðw2 vÞðv w1 Þ: ð34Þ Throughout we will set w2 = 1. When harvesting is included, the layer equation (26) becomes v00 þ cv0 þ vðgðv þ U Þ bÞ where bv is the harvesting term. Since GðvÞ ¼ vðgðv þ U Þ bÞ is a cubic function of v one can explicitly find the wave speed c. Indeed, since GðvÞ ¼ vðr2 vÞðv r1 Þ; where we have ordered the roots 0 < r1 < r2. It follows from [20] that pffiffiffir2 r1 c ¼ 2 2 ð35Þ Fig. 5. Numerical simulations with w2 = 1, w1 = 1/2, z0 = 1/5, c11 = 1, and b = 0. (a) Climax field and (b) pioneer field. 32 S. Brown et al. / Mathematical Biosciences 194 (2005) 21–36 Fig. 6. Numerical simulations with w2 = 1, w1 = 1/2, z0 = 1/5, c11 = 1, and b = 4/100. (a) Climax field and (b) pioneer field. and recall this is the leading order approximation to the wave speed. Note that the roots ri, i = 1,2 depend on U and b. U is the root of F given in (24) that lies in the interval (0, z0/c11). Using the formulas (24) and (35) we find that for w2 = 1, w1 = 1/2, c11 = 1 and z0 = 1/5 that c > 0 at b = 0 while c < 0 when b = 4/100. Numerical solutions to the full system (12) and (13) on a one dimensional domain are shown in Figs. 5 and 6. For the simulation shown in Fig. 5, b = 0 and initial data was a scaled Heaviside function for both v and u. We see that this initial data quickly evolves into a traveling wave with positive wave speed. The climax species eventually dominates the entire one dimensional domain while the pioneer species declines. Such a wave with c > 0 represents an extinction wave for the pioneer species. Fig. 7. Numerical simulations with w2 = 1, w1 = 0.3, z0 = 0.2, c11 = 1, and b = 0. (a) The v = 1/2 level sets at several different times. (b) Final v surface plot. S. Brown et al. / Mathematical Biosciences 194 (2005) 21–36 33 Fig. 8. Numerical simulations with w2 = 1, w1 = 0.3, z0 = 0.2, c11 = 1, and b = 12/100. (a) The v = 1/2 level sets at several different times. (b) Final v surface plot. For the simulation displayed in Fig. 6 we use as initial data the final values of u and v from Fig. 5 and simply change b from zero to b = 4/100. There we see that the wave speed is now negative and the pioneer species is spreading over the domain while the climax species is receding. Similar results are obtained by changing the parameter c11. With w1 = 1/2, for any c11 > c the zeroth order wave speed is always positive. However, when w1 = 2/5 the wave speed is negative if c11 is large enough resulting in the pioneer species invading the domain. We have also carried out numerical simulations on two dimensional domains with periodic boundary conditions. With b = 0 and initial conditions chosen randomly the solution to (12) and (13) evolves into a pattern where the climax species invades the domain. In Fig. 7(a) we show the v = 1/2 level sets of the solution at several different times. The arrows indicate the direction that the level sets move with increasing time. We see that the wave is locally a plane wave whose wave speed depends on local curvature effects. Such curvature effects are generally expected for this type of model (see [21]). In Fig. 7(b) we have displayed the climax field v at the end of the numerical simulation. Lastly, in Fig. 8 we set the harvesting parameter b = 12/100 and use the final values of the v and u fields from Fig. 7 as initial conditions for the system. As before, the v = 1/2 level sets are plotted at several different times in Fig. 8(a). It is clear that the climax species is now receding. In particular, the plane wave speed has changed sign and the pioneer species is now invading the domain. The final v field is shown in Fig. 8(b). 5. Discussion In this paper we have examined a system of reaction-diffusion equations modeling the interaction of pioneer and climax species. We have shown traveling waves exist when the climax species diffusion rate is small relative to that of the pioneer species. Numerical results indicate that these waves are in fact stable. Rigorous stability results follow from the work of ([14]). Moreover, we 34 S. Brown et al. / Mathematical Biosciences 194 (2005) 21–36 Fig. 9. Numerical simulations with w2 = 1, w1 = 1/2, z0 = 0.4, c11 = 1/2, and D1 = D2. demonstrated how the leading order approximation to the wave speed can be used to study how various model parameters effect the sign of the wave speed. This is important since the sign of the wave speed determines which species becomes extinct. In our concluding remarks we mention that the same model has a rich set of other dynamic behaviors. For instance, with the pioneer and climax diffusivities equal and w2 = 1 we have found numerically traveling waves that connect the rest state (0,1) with various interior rest states. For the two simulations shown in Fig. 9 the parameters are set so that we are in Case 2 (see Fig. 2). In Fig. 10. Numerical simulations with w2 = 1, w1 = 1/2, c11 = 0.27, and b = 0.045. (a) Climax field and (b) pioneer field. S. Brown et al. / Mathematical Biosciences 194 (2005) 21–36 35 this case, (8) and (9) has a stable interior rest state. With equal diffusion coefficients, the analysis in section 3 does not apply. Intuitively, however, we except that if there is a traveling waves connecting (0,1) to a stable interior rest state that leaves the stable state (0,1) in its wake, then by increasing the harvesting of the pioneer one should be able to reverse the speed of the wave. In Fig. 9 we show numerically generated traveling waves at two different values of b, the harvesting parameter. There we show that by increasing the harvesting parameter we can indeed see a change in the sign of the wave speed. As was indicated in Section 2, in Case 2 the stable interior equilibria can undergo a Hopf bifurcation resulting in the existence of stable periodic orbits for the reaction system (8) and (9), see e.g. [5]. Using these parameter values for the reaction-diffusion system with equal diffusivities, we have found numerically point to periodic traveling waves. These are waves that connect a stable periodic orbit to a stable rest state of the reaction system. At the parameter values given in Fig. 10 the reaction system has both a stable periodic orbit and a stable rest state at (0,1). In Fig. 10 we have displayed a numerical solution to the reaction-diffusion system when these parameter values are used. There we see a point to periodic traveling wave which is leaving in its wake a stable periodic orbit. It should be noted that similar types of waves have been shown to exist in predator-prey systems (see [22]). Lastly we note that extremely complicated spatio-temporal dynamics can be generated at other parameter values. In particular, we have found numerical solutions which seem to arise in the manner discussed in [23]. At present we are investigating these as well as standing patterns. Acknowledgement This work was supported by the National Science Foundation Grants DMS-94-04-521 and DMS-94-04-60. References [1] W.E. Ricker, Stock and recruitment, J. Fish. Res. Bd. Can. 11 (1954) 559. [2] M.P. Hassell, H.N. Comins, Discrete time models for two-species competition, Theoret. Population Biol. (9) (1976) 202. [3] J.M. Cushing, Nonlinear Matrix Models and Population Dynamics, Nat. Resour. Model 2 (1988) 539. [4] J.F. Selgrade, G. Namkoong, Stable periodic behavior in a pioneer-climax model, Nat. Resour. Model 4 (1990) 215. [5] J.F. Selgrade, G. Namkoong, Population interactions with growth rates dependent on weighted densities, in: S. Busenberg, M. Martelli (Eds.), Differential Equation Models in Biology, Epidemiology and Ecology, Lecture Notes in Biomath, 92, 1991, p. 247. [6] J.F. Selgrade, Planting and harvesting for pioneer-climax models, Rocky Mountain J. Math. 24 (1994) 293. [7] J. Robert Buchanan, Asymptotic Behavior of Two Interacting Pioneer/Climax Species, Fields Inst. Comm. (21) (1999) 51. [8] Sumner Suzanne, Hopf bifurcation in competing species models with linear and quadratic fitnessesProceedings of Dynamic Systems and Applications 2, Vol. 2, Dynamic Publishers, Inc, 1996, p. 535. [9] S. Sumner, Hopf Bifurcation in Pioneer-Climax Competing Species Models, Math. Biosci. 137 (1996) 1. 36 S. Brown et al. / Mathematical Biosciences 194 (2005) 21–36 [10] M.R. Owen, M.A. Lewis, How Predation can Slow, Stop or Reverse a Prey Invasion, Bull. Math. Biol. 63 (2001) 655. [11] J.F. Selgrade, J.H. Roberds, Lumped-density population models of pioneer-climax type and stability analysis of Hopf bifurcations, Math. Biosci. 135 (1996) 1. [12] C. Neuhauser, Mathematical Challenges in spatial Ecology, Notices of the AMS 45 (11) (2001) 1304. [13] D. Tilman, P. Kareiva, Spatial Ecology: The Role of Space in Population Dynamics and Interspecific Interactions, Princeton University, 1997. [14] R. Gardner, C.K.R.T. Jones, Stability of travelling wave solutions of diffusive predator-prey systems, Trans. AMS 327 (1991) 465. [15] P.C. Fife, Lecture Notes in Biomathematics: Mathematical Aspects of Reacting and Diffusing Systems, Springer, New York, NY, 1979. [16] C.K.R.T. Jones, Geometric singular perturbation theory, C.I.M.E. Lectures, Session on Dynamical Systems 1994, p. 1. [17] P. Szmolyan, Transversal heteroclinic and homoclinic orbits in singular perturbation problems, J. Differential. Equations 92 (1991) 252. [18] N. Fenichel, Geometric Singular Perturbation Theory for Ordinary Differential Equations, J. Differential Equations 31 (1979) 53. [19] S.L. Brown, A Reaction Diffusion Model for Competing Pioneer and Climax species Ph.D. Thesis, Montana State University, Bozeman, MT, 1998. [20] R. Casten, H. Cohen, P. Lagerstrom, Perturbation analysis of an approximation to the Hodgkin-Huxley Theory, Quart. Appl. Math. 111 (32) (1975) 365. [21] P. Fife, Dynamics of Internal Layers and Diffusive Interfaces, SIAM, 1988. [22] A. Dunbar, R. Steve, Traveling Waves in Diffusive Predator-Prey Equations: Periodic Orbits and Point-ToPeriodic Heteroclinic Orbits, SIAM J. Appl. Math 46 (6) (1996). [23] J.H. Merkin, V. Petrov, S.K. Scott, K. Showalter, Wave induced chaos in a continuously-fed unstirred reactor, J. Chem. Soc. Faraday Transactions 92 (16) (1996) 2911.