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Mathematical Biosciences 194 (2005) 21–36
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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:
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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.
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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 .
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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].
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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
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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.
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