Bulletin of Mathematical Biology 67 (2005) 637–661
www.elsevier.com/locate/ybulm
Regimes of biological invasion in a predator–prey
system with the Allee effect
Sergei Petrovskiia,b,∗, Andrew Morozova,b, Bai-Lian Lib
a Shirshov Institute of Oceanology, Russian Academy of Science, Nakhimovsky Prospekt 36,
Moscow 117218, Russia
b Ecological Complexity and Modeling Laboratory, Department of Botany and Plant Sciences,
University of California at Riverside, Riverside, CA 92521-0124, USA
Received 28 April 2004; accepted 15 September 2004
Abstract
Spatiotemporal dynamics of a predator–prey system is considered under the assumption that prey
growth is damped by the strong Allee effect. Mathematically, the model consists of two coupled
diffusion-reaction equations. The initial conditions are described by functions of finite support which
corresponds to invasion of exotic species. By means of extensive numerical simulations, we identify
the main scenarios of the system dynamics as related to biological invasion. We construct the maps
in the parameter space of the system with different domains corresponding to different invasion
regimes and show that the impact of the Allee effect essentially increases the system spatiotemporal
complexity. In particular, we show that, as a result of the interplay between the Allee effect and
predation, successful establishment of exotic species may not necessarily lead to geographical spread
and geographical spread does not always enhance regional persistence of invading species.
© 2004 Society for Mathematical Biology. Published by Elsevier Ltd. All rights reserved.
1. Introduction
Biological invasion has been attracting considerable attention recently due to its
numerous adverse effects on ecosystem dynamics and biodiversity (Hengeveld, 1989;
∗ Corresponding author at: Shirshov Institute of Oceanology, Russian Academy of Science, Nakhimovsky
Prospekt 36, Moscow 117218, Russia.
E-mail address: [email protected] (S. Petrovskii).
0092-8240/$30 © 2004 Society for Mathematical Biology. Published by Elsevier Ltd. All rights reserved.
doi:10.1016/j.bulm.2004.09.003
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Hastings, 1996; Shigesada and Kawasaki, 1997; Frantzen and van den Bosch, 2000; Keitt
et al., 2001; Owen and Lewis, 2001; Wang and Kot, 2001). Although a considerable
progress has been made during the last decade in understanding basic scenarios of species
invasion, many important issues have not been properly addressed yet. Comprehensive
identification of factors that affect rates of invasion and patterns of species spread and can
potentially either enhance or hamper species invasion, is expected to open a possibility
of biological control and to result in effective invasive species management (Sakai et al.,
2001; Fagan et al., 2002).
Biological invasion is known to have a few more or less clearly distinguishable stages
[cf. Shigesada and Kawasaki (1997), Sakai et al. (2001)]. The first stage is introduction
when a few organisms of an exotic species are brought, deliberately or unintentionally,
into the given ecosystem. The second stage is establishment when the introduced species is
getting adapted to the new environmental conditions. The third stage is, in case the previous
two have been successful and did not result in species extinction, the ‘geographical’ spread
when the exotic species invades new areas at the scale much larger compared to the domain
where it was originally introduced. Later stages are related to the impact of the new species
on the native ecological community and, possibly, on human health and society.
Apparently, each stage has its own basic processes and specific problems. In this paper,
we mainly focus on the spatiotemporal dynamics of the introduced species typical for
the second and third stages of invasion. Thus, we are mainly interested in such issues as
population growth, species extinction/persistence, patterns of species spread and related
ecological pattern formation. Under what conditions the introduced species will fail to
establish itself in the new environment, may it happen that successful introduction will
not lead to geographical invasion, whether the spatial spread will take place through
propagation of the population front or in a more complicated manner—all these questions
are highly relevant both from a theoretical point of view and from the point of immediate
practical applications.
From a theoretical perspective, it is well-known that many basic features of the
species spread during biological invasion can be explained reasonably well by the
interplay between local population growth and local dispersal due to self-motion of
individuals (Fisher, 1937; Skellam, 1951; Okubo, 1980; Shigesada and Kawasaki, 1997).
Mathematically, this model is described by a diffusion-reaction equation whose properties
appear to depend essentially on the type of the population growth. While early studies
tended to assume it to be logistic, more recently much attention has been paid to the
impact of the Allee effect (Lewis and Kareiva, 1993; Owen and Lewis, 2001; Wang and
Kot, 2001) because the Allee effect was shown to affect virtually all aspects of species
interactions in space and time (Allee, 1938; Berryman, 1981; Dennis, 1989; Amarasekare,
1998; Courchamp et al., 1999; Gyllenberg et al., 1999).
The Allee effect usually arises as a result of intraspecific interactions (Allee, 1938;
Berryman, 1981). However, the impact of interspecific interactions on species invasion is
important as well. In particular, it was shown that predation is likely to affect the rates of
invasive species spread (Fagan and Bishop, 2000; Owen and Lewis, 2001; Petrovskii et al.,
in press). The spatiotemporal dynamics of a predator–prey system relevant to biological
invasion has been recently studied in much detail in the case that population growth is
logistic (Petrovskii et al., 1998; Petrovskii and Malchow, 2000). However, the impact of the
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Allee effect has not been properly addressed, although it was shown that it can significantly
increase the complexity of the system dynamics (Petrovskii et al., 2002a,b; Morozov, 2003;
Morozov et al., 2004).
In this paper we consider a predator–prey system where prey growth is damped by the
Allee effect. By means of extensive computer simulations, we fulfil a thorough study of this
system in connection to biological invasion and give a detailed classification of possible
patterns of species spread. We show that the system dynamics is remarkably rich and that
its complexity increases with an increase of the prey maximum growth rate. In particular,
we show that, for sufficiently large prey growth rate, there is a parameter range where the
pattern of species spread exhibits nonuniqueness subject to the initial conditions. We also
show that, as a result of the interplay between predation and the Allee effect, successful
establishment of an exotic species does not necessarily lead to its geographical spread
and that geographical spread, if/when it takes place, does not guarantee species regional
persistence.
2. Main equations
We consider the following 1-D model of predator–prey interaction in a homogeneous
environment:
∂ H (X, T )
∂2 H
= D1
+ F(H ) − f (H, P),
(1)
∂T
∂ X2
∂2 P
∂ P(X, T )
= D2 2 + κ f (H, P) − M P
(2)
∂T
∂X
[cf. Nisbet and Gurney (1982), Murray (1989), Holmes et al. (1994), Sherratt (2001)]. Here
H and P are the densities of prey and predator, respectively, at moment T and position X.
D1 and D2 are diffusivities and κ is the food utilization coefficient. The function F(H )
describes prey multiplication, f (H, P) describes predation, and the term M P stands for
predator mortality.
We consider Holling type II response for predator and use the following
parametrization:
AH P
(3)
f (H, P) =
H+B
where A describes predation intensity and B is the half-saturation prey density.
We assume that prey population is damped by the Allee effect, its growth rate being
parametrized as follows (Lewis and Kareiva, 1993):
4ω
F(H ) =
(4)
H (H − H0)(K − H )
(K − H0)2
where K is the prey carrying capacity, ω is the maximum per capita growth rate and H0
quantifies the intensity of the Allee effect so that it is called ‘strong’ if 0 < H0 < K
(when the growth rate becomes negative for H < H0 ) and ‘weak’ if −K < H0 ≤ 0 [cf.
Owen and Lewis (2001), Wang and Kot (2001)]. For H0 ≤ −K , the Allee effect is absent
(Lewis and Kareiva, 1993).
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For convenience, we introduce dimensionless variables u = H /K , v = P/(κ K ),
t = aT , x = X (a/D1 )1/2 where a = Aκ K /B. Then, from Eqs. (1) and (2), we obtain:
uv
∂ 2u
∂u(x, t)
+ γ u(u − β)(1 − u) −
=
,
∂t
1 + αu
∂x2
∂v(x, t)
∂ 2v
uv
= 2+
− δv.
∂t
1 + αu
∂x
(5)
(6)
Eqs. (5) and (6) contain five dimensionless parameters (against nine in the original
equations), i.e., α = K /B, β = H0 /K , γ = 4ωB K /(Aκ(K − H0)2 ), δ = M/a and
= D2 /D1 . Thus, the behaviour of dimensionless solutions u and v appears to depend
on five dimensionless combinations of the original parameters rather than on each of them
separately.
Invasion of an alien species is usually started when a number of individuals of an exotic
species is locally brought into the given ecosystem. From the point of model (5) and (6), it
means that the initial species distribution should be described by functions of finite support.
Thus, we consider initial conditions of the following form:
u(x, 0) = u 0
for −∆u < x < ∆u ,
otherwise u(x, 0) = 0,
(7)
v(x, 0) = v0
for −∆v < x < ∆v ,
otherwise v(x, 0) = 0
(8)
where u 0 , v0 are the initial population densities and ∆u , ∆v give the radius of the initially
invaded domain. Initial conditions (7) and (8) also correspond to the problem of biological
control when, soon enough after introduction of an exotic species, a predatory species is
introduced intentionally in an attempt to slow down or stop its spread [cf. Fagan and Bishop
(2000), Owen and Lewis (2001), Petrovskii et al. (in press)].
Note that the initial conditions (7) and (8) are somewhat idealized and in reality the form
of the species initial distribution can be much more complicated. However, the results of
our computer simulations show that the type of the system dynamics depends more on the
radius of the initially inhabited domain and on the population density inside rather than on
the details of the population density profile.
3. Patterns of species spread
From the point of ecological applications, it is very important to distinguish between
the cases when invasion will likely be successful and the cases when it will likely fail.
In practical ecology, invasion failure usually means that the introduced species fails to
establish itself in the new environment. Thus, an unsuccessful species is expected to go
extinct soon after its introduction. However, it remains unclear whether invasion failure
may happen as well at later stages, i.e., whether invasive species can go extinct after having
already spread over relatively large areas. The results that we present in this section show
that, for an invasive species affected by the Allee effect, extinction may take place at a later
stage after its geographical spread.
Moreover, it seems reasonable to distinguish between ‘geographical’ invasion and
‘local’ invasion. We will call the invasion ‘local’ in the case when the new species
successfully establishes itself locally, i.e., around the place of original introduction, but
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641
does not spread over new areas due to the impact of certain factors. Correspondingly,
we call the invasion ‘geographical’ in the case that the invasive species succeeds to spread
over large areas. In many cases, local invasion of exotic species is followed by geographical
invasion, although the time lag between these two stages can be as long as a few decades.
The question that still needs to be answered is what environmental or biological factors
make this lag so long and whether local invasion must always be followed, sooner or later,
by geographical invasion. The results that we present below show that an exotic species can
invade locally but sometimes fails to invade geographically due to the interplay between
predation and the Allee effect.
Eqs. (5) and (6) with the initial conditions (7) and (8) were solved numerically in the
domain −L < x < L by finite-difference method. The steps of the numerical mesh were
chosen as x = 0.2 and t = 0.001 and it was checked that a decrease of the mesh steps
did not lead to any significant modification of the results. The ‘no-flux’ condition was used
at the boundaries and the radius L of the numerical domain was chosen large enough in
order to make the impact of the boundaries as small as possible during the simulation time.
Throughout this section we fix = 1, the effect of differential diffusivity will be addressed
in Section 5.
In order to identify different regimes and to reveal the corresponding structure of the
parameter space, in total, over three thousand computer experiments were run for different
parameter values. We obtain that all regimes observed in our simulations can be classified
into three groups, see Fig. 1. These groups correspond to extinction, geographical invasion
when the species keep spreading until they reach the domain boundaries, and regional
persistence when the alien species invade locally and spread over a certain area but do not
go farther.
Before proceeding to regime description, the issue of regime dependence on the initial
conditions should be clarified. It is well-known that in a predator–prey system with logistic
growth for prey, although the population density can fall to very small values, the species
will never go extinct in the strict mathematical sense because the extinction state (0, 0) is
unstable (Gilpin, 1972) and thus acts as a ‘repeller’. Evolution of the initial conditions (7)
and (8) eventually leads, for any biologically reasonable parameter values, to formation of
a travelling population front. Thus, the large-time asymptotical system dynamics does not
depend on the initial conditions as long as they are described by finite functions (Volpert
et al., 1994). Although the actual patterns of spread can be different for different parameter
values, e.g., front propagation can be followed either by a steady spatially homogeneous
species distribution or by spatiotemporal pattern formation in the wake (Sherratt et al.,
1995; Petrovskii et al., 1998; Petrovskii and Malchow, 2000), any introduction of a new
species will lead to its geographical invasion. This prediction of inevitable species spread
does not seem realistic and was used as a justification for various modifications of the
model, e.g., by implementing a threshold at low population densities (Brauer and Soudack,
1978; Wilson, 1998; Petrovskii and Shigesada, 2001).
The situation becomes different when the invasive prey is affected by the Allee effect.
In this case, already single-species models predict that the introduced species does go
extinct when the population size is not large enough (Lewis and Kareiva, 1993; Petrovskii,
1994; Petrovskii and Shigesada, 2001). [Note that this phenomenon is often observed in
nature as well, see Courchamp et al. (1999).] The impact of predation makes this threshold
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Fig. 1. Classification of invasion regimes.
behaviour more prominent, cf. Section 3.2. Essentially, for any value of parameters in
Eqs. (5) and (6), sufficiently small u 0 and/or ∆u can turn any regime of species spread to
extinction. In order to exclude this somewhat trivial case, in our computer experiments u 0
and ∆u are always chosen sufficiently large.
3.1. Regimes of geographical invasion
We begin with the regimes describing the unbounded spread of the invasive species
which corresponds to the geographical stage of biological invasion. We found three
different scenarios of the species spread, examples are shown in Figs. 2–8.
According to the first scenario, the species is spreading over space through propagation
of a travelling population front, see Figs. 2–5. In front of the front the species is absent,
behind the front it is present in considerable densities. Apparently, this type of species
spread corresponds to successful invasion. Depending on parameter values, in the wake of
the front there can arise either a stationary spatially homogeneous species distribution or
irregular spatiotemporal population oscillations.
Fig. 2 shows the snapshots of the population density (solid curve stands for prey, dashed
for predator) obtained for parameters α = 0.5, β = 0.27, γ = 3, δ = 0.51. Here
and below (except for Fig. 5), the initial conditions are ∆u = 7, ∆v = 2, u 0 = 1,
v0 = 0.1. Propagation of the population front is followed by a stationary homogeneous
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Fig. 2. Snapshots of the population density showing species invasion through propagation of population front
with steady homogeneous species distribution in the wake, parameters are given in the text. Here and below, solid
curve stands for prey, dashed for predator.
species distribution with the population density corresponding to the stable steady state
of the homogeneous system. (For some other parameter values, the front can be followed
by a succession of a few promptly damping oscillations preceding the region of spatial
homogeneity.) It is this pattern of species spread that is usually evoked in connection with
biological invasion described by diffusion-reaction equations; moreover, for a long time it
had been considered as the only possible regime that deterministic diffusive predator–prey
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Fig. 3. Snapshots of the population density showing species invasion through propagation of population front
with irregular spatiotemporal oscillations in the wake in the case that the amplitude of oscillation is not large.
systems can provide [cf. Lewis (1996)]. For the parameters of Fig. 2, the population fronts
of prey and predator propagate with the same speed. For other parameter values it may
happen that the front of prey travels with a greater speed than the front of predator (see
also Section 4). In that case, the predator invades into the space already inhabited by prey
at its carrying capacity.
However, for the parameter values when the homogenous steady state becomes unstable,
the pattern of spread changes essentially. Fig. 3 shows the snapshots of the population
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645
Fig. 4. A regime of invasion similar to the one shown in Fig. 3 but in the case that the amplitude of spatiotemporal
species oscillations is large.
density obtained for parameters α = 0.5, β = 0.27, γ = 3, δ = 0.485. In this case,
propagation of the population front is followed by excitation of irregular spatiotemporal
oscillations in population density. A similar phenomenon was observed earlier for the
diffusive predator–prey system with logistic growth [cf. Sherratt et al. (1995)]. Note that
the domain with irregular oscillations is separated from the travelling front by a ‘plateau’,
i.e., by a domain with nearly-homogeneous species spatial distribution. The values of
the population density in this plateau correspond to the locally unstable equilibrium.
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Fig. 5. Species invasion through propagation of a periodic population wave which is generated by a spatially
irregular ‘wave-maker’ situated around the place of species introduction.
This phenomenon of ‘dynamical stabilization’ was considered in detail in Petrovskii and
Malchow (2000), Petrovskii et al. (2001) and Malchow and Petrovskii (2002).
The results of our computer experiments show that the pattern of spread when
propagation of the population front is followed by excitation of irregular spatiotemporal
oscillations is rather typical for the diffusive predator–prey system with the Allee effect in
the sense that it can be observed in a wide parameter range. Some features of the regime
can vary with parameter values; for instance, the unstable plateau does not always exist.
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647
Fig. 6. Snapshots of the population density (only half of the domain is shown) showing species spread over space
through propagation of a solitary moving population patch, or a ‘pulse’. Note that this pattern of spread does not
lead to species invasion because the population density in the wake is zero.
Also, for some parameters the amplitude of population oscillations becomes notably larger
so that the pattern in the wake becomes more prominent, see Fig. 4 obtained for α = 0.5,
β = 0.27, γ = 3, δ = 0.47. Note that, although in this case the pattern as a whole looks
like an ensemble of separated patches, the scenario of species invasion is still essentially
the same: close inspection of the population density snapshots clearly reveals the travelling
population front, cf. top, middle and bottom of Fig. 4.
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Fig. 7. A regime similar to that shown in Fig. 6 but in the case that the shape of the wave exhibits oscillations.
Only half of the domain is shown.
It should be mentioned that for this regime, as well as for propagation of smooth
population fronts, it is possible that for some parameter values the front of prey travels
faster than the front of predator, see Section 4 for more details. In that case, first, invasion
of prey takes place. The travelling population front of prey separates the domain where
both species are absent (in front of the front) from the domain where prey is at its
carrying capacity and predator is absent (behind the front). The population front of predator
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Fig. 8. A regime of invasion through propagation of separate patches and groups of patches.
propagates into the region already inhabited by prey, in the wake of the predator front
irregular population oscillations arise.
In the cases shown in Figs. 3 and 4, the patches arising behind the front are irregular.
For other parameter values, however, the spatial structure can be more regular. Fig. 5 shows
the snapshots of the population density obtained for parameters α = 0.5, β = 0.28, γ = 7,
δ = 0.46 and the initial conditions ∆u = 5, ∆v = 3, u 0 = 1, v0 = 1. In this case, species
invasion takes place through propagation of a periodic travelling wave. The periodic wave
is generated by an irregular ‘wave-maker’ situated about the place of the initial species
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distribution, i.e., around x = 0. Depending on parameter values, the wave-maker either
stays localized or gradually grows in size. In the latter case, the periodic oscillations are
eventually displaced by irregular ones [cf. Petrovskii and Malchow (2001) for a similar
phenomenon].
All the regimes described above correspond to species invasion through propagation of
a population front. According to the second scenario, the species spread over the domain
via propagation of a moving patch, or ‘pulse’, cf. Figs. 6 and 7 (only a half of the domain is
shown). The dynamics is similar to predator–prey pursuit [cf. Murray (1989)]. In this case,
the invasive species is absent both in front of the pulse and in its wake which apparently
means that invasion fails in spite of the fact that geographical spread has taken place.
Depending on parameter values, the travelling population pulse can be either stationary
when its shape does not change with time, or nonstationary when its shape oscillates with
time; in both cases the pulse propagates with a constant speed. Fig. 6 shows the snapshots
of the population density obtained for parameters α = 0.5, β = 0.28, γ = 3, δ = 0.44
when the species spread over the system via propagation of a stationary travelling pulse.
Fig. 7 is obtained for α = 0.5, β = 0.28, γ = 3, δ = 0.425, it shows propagation of a
nonstationary pulse.
Finally, there is another, more exotic scenario of species geographical spread. Fig. 8
shows the snapshots of the population density obtained for parameters α = 0.05, β = 0.28,
γ = 3, δ = 0.52. In this case, invasion takes place through formation and propagation of
groups of moving patches. However, the patch motion is now much more complicated than
the simple locomotion in the case of travelling pulses. The patches interact with each other,
they merge and split, some of the patches or even groups of patches can disappear, new
patches are formed, they can produce new groups of patches, etc. The inhabited area grows
and eventually the groups of nonstationary patches occupy the whole domain. Although
there is certain visual similarity between this regime of invasion and the regime shown
in Fig. 4, comparison between the snapshots obtained for different times immediately
shows that in this case there is no stationary travelling population front. Another important
distinction is that the size of the domain occupied by the moving patches does not grow
monotonically, cf. top, middle and bottom of Fig. 8. That happens when the leading group
of patches goes extinct in the course of the system dynamics.
3.2. Regimes of anomalous extinction
The second group of regimes corresponds to species extinction. It is well-known that,
in case the introduced species is affected by the Allee effect, it goes extinct if the initial
population size is not large enough, i.e., either the radius of originally inhabited domain
or the population density inside are less than certain critical values (Lewis and Kareiva,
1993; Petrovskii, 1994; Petrovskii and Shigesada, 2001). In this case, the population size
decreases exponentially and the population stays localized in about the same domain where
it had originally been introduced. We will refer to this type of population dynamics as
‘ordinary’ extinction.
When the introduced population is affected by predation, ‘ordinary’ extinction can
takes place as well. Although the critical values for the initial radius and the initial
prey density appear somewhat larger in this case as a result of the pressure from
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the predator, the qualitative features of ‘ordinary’ extinction remain the same, cf.
the previous paragraph. (Note that, in our model, extinction of prey inevitably leads
to extinction of both species.) However, due to the impact of predation, species
extinction can also follow other, rather unusual scenarios. Depending on parameter
values, we observed two other regimes when species extinction is either preceded by
formation of a distinct long-living spatiotemporal pattern or by long-distance population
spread.
Fig. 9 shows the snapshots of the population density in the regime of species extinction
through ‘dynamical localization’ observed for parameters α = 0.1, β = 0.15, γ = 5,
δ = 0.257. At an early stage of the system dynamics, a moving patch is formed which
propagates with approximately constant speed over distances much longer compared to the
radius of the initial species distribution. This stage of the system dynamics is apparently
similar to pulse propagation, see Fig. 6. Finally, however, the prey is caught by the predator
and both species go extinct: starting from the moment shown in the bottom of Fig. 9 the
population size exhibits fast decay.
Fig. 10 shows the regime of ‘patchy extinction’ (observed for α = 0.5, β = 0.2, γ = 1,
δ = 0.365). In this case, the initial conditions eventually evolve into an ensemble of patches
allocated over the domain. The patches interact with each other in a complicated manner
somewhat similar to the patch dynamics shown in Fig. 8; finally, however, the species go
extinct.
We want to emphasize that, in both of these cases, the invasive population persists during
a remarkably long time before the actual population decay takes place (for the parameters
of Figs. 9 and 10, nearly one hundred times longer than it would be in the case of the
‘ordinary’ extinction) and it can spread over large distances. During that period, the system
dynamics is very similar to the regimes of geographical spread shown in Figs. 6 and 8,
respectively. These results seem to reveal a new aspect of the ‘extinction debt’ (Tilman
et al., 1994; Loehle and Li, 1996) and also evoke a more general discussion regarding the
ecological relevance of transient dynamics (Hastings, 2001): a population which is doomed
to vanish, e.g., as a result of certain unfavourable environmental changes, can exhibit the
dynamics which is, during a long time, virtually indistinguishable from the dynamics of
persistent populations. The collapse comes unexpectedly and then it may be too late to
apply a conservation strategy.
3.3. Regimes of local invasion
Remarkably, the two groups of the regimes described above, i.e., species extinction and
unbounded spatial spread, do not exhaust all possible types of the system dynamics. For
certain parameter values, evolution of the initial species distribution leads to formation of
quasistationary patches. Fig. 11 shows the snapshots obtained for α = 0.5, β = 0.32,
γ = 3, δ = 0.455. In this case, at an early stage of the system dynamics (for t 100), two
symmetric dome-shaped patches are formed. At later stages, the position of their centres
remains fixed and the shape of the patches changes with time in an oscillatory manner.
A close inspection shows that, depending on the parameter values, the corresponding
temporal fluctuations in the population density can be either periodic or chaotic (Morozov
et al., 2004).
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Fig. 9. An example of ‘dynamical localization’: the population densities form a solitary moving patch which
propagates a long distance before the species go extinct. Only half of the domain is shown.
From the point of prospective ecological applications, this regime of the system
dynamics is probably the most interesting. Field observations give many examples when
the invasive species, after introduction, remains localized inside a certain area during a
long time. Thus, their local invasion and subsequent regional persistence is not followed
by geographical spread. There exist different explanations of this phenomenon such as the
impact of environmental borders, time-lag related to mutations and evolutionary changes
caused by adaptation in the new environment, etc. Our results provide another explanation
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Fig. 10. An example of long-living transients corresponding to the ‘patchy extinction’: species extinction is
preceded by spatiotemporal pattern formation going during a relatively long time.
and show that invasive species can be held localized purely due to certain inter- and
intraspecific interactions such as the interplay between the Allee effect and predation.
4. Parameter space structure
In the previous section, we demonstrated that the predator–prey system with the Allee
effect for prey exhibits very rich dynamics and predicts a wide variety of patterns/regimes
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Fig. 11. The regime of local invasion when the initial species introduction leads to formation of a few standing
patches. The position of the patches does not change with time while their shape is either stationary or oscillates
with time depending on parameter values.
of species spread. A natural question arising here is about the relation between the
regimes and possible transition between them that may take place in response to parameter
variation. One way to address this issue is to study the structure of the system parameter
space in order to locate the domains corresponding to different regimes of invasion.
For that purpose, we fulfil a detailed numerical study of the system dynamics. In total,
about three thousand computer experiments were run for different parameter values. The
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Fig. 12. The map in the parameter plane of the Eqs. (5) and (6) for γ = 1. Different domains correspond to
different regimes, see details in the text.
results are shown in Figs. 12–14. Note that, since the model depends on four parameters
(assuming here that = 1 is fixed, the case = 1 will be addressed separately in the
next section), it looks virtually impossible to accomplish a detailed study of the whole
R4+ parameter space. It is readily seen that in this case even as many as 104 computer
experiments, if run for parameter values spread homogeneously over the parameter space,
would provide only meagre information about the position of different domains. In order
to overcome this difficulty, we apply a certain strategy in choosing parameter values. Since
the main goal of this paper is to study the dynamics of invasive species subject to the
interplay between the impact of the Allee effect (quantified by β) and predation (quantified
by δ), it looks more relevant to focus on the detailed structure of the (δ, β) plane.
Correspondingly, we firstly choose a certain hypothetical value for the half-saturation
density, i.e., α = 0.5. Then we select a few values of the maximum per capita prey growth
rate, i.e., in dimensionless units, γ = 1 (Fig. 12), γ = 3 (Fig. 13) and γ = 7 (Fig. 14).
Then, for each of these values, the (δ, β) parameter plane was studied thoroughly.
In order to make the search in the (δ, β) plane more effective, one should also take into
account that nontrivial dynamics can only take place inside the rectangle {0 < δ ≤ Ω =
(1 + α)−1 , 0 < β ≤ 0.5}. Here δ is positive due to its biological meaning and β is assumed
to be positive because we are concerned with consequences of the strong Allee effect. The
values δ > Ω = (1 + α)−1 correspond to predator extinction because the phase plane of
Eqs. (5) and (6) in the spatially homogeneous case does not possess a co-existence steady
state in the biologically meaningful domain {u ≥ 0, v ≥ 0} and the ‘prey-only’ steady
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Fig. 13. The map in the parameter plane of the Eqs. (5) and (6) for γ = 3.
state (1, 0) is a stable node. The values β > 0.5 are readily seen to always correspond to
the ‘ordinary extinction’ and thus are not of much interest either.
Fig. 12 shows the map in the (δ, β) plane obtained for γ = 1. Here domain 1
corresponds to species extinction (including both ‘ordinary’ and ‘anomalous’ extinction).
Domain 3 corresponds to geographical spread of invasive species either through
propagation of population fronts with irregular spatiotemporal oscillation in the wake
(Figs. 3 and 4) or through the spread of moving patches (Fig. 8). Since it is not always
easy to distinguish between these two regimes in numerical experiments, we are not going
into the details of the ‘fine structure’ of domain 3. Domain 4 corresponds to geographical
invasion through propagation of smooth population fronts with stationary homogeneous
species distribution in the wake (Fig. 2). Sub-domain 4∗ corresponds to the case when the
front of invasive prey travels faster than the front of invasive predator, here the dashed
curve separating sub-domain 4∗ can be obtained analytically [cf. Petrovskii and Malchow
(2000)]. Domain 5 corresponds to local invasion through formation of quasi-stationary
patches, see Fig. 11.
The structure of the (δ, β) parameter plane changes for larger values of γ . Fig. 13 shows
the map obtained for γ = 3, notations are the same as above. Now a new domain appears,
i.e., domain 2 corresponding to propagation of solitary population pulses. Sub-domain 3∗
corresponds to the case when the front of prey travels faster than the front of predator, cf.
Section 3.1 for details. A higher value of γ corresponds to a ‘stronger’ prey; thus, it is not
surprising that the domain where the front of prey outruns the front of predator (below the
dashed curve) has grown in size compared to the case γ = 1 shown in Fig. 12.
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657
Fig. 14. The map in the parameter plane of the Eqs. (5) and (6) for γ = 7. Domain 6 corresponds to nonuniqueness
of invasion regime, see details in the text.
Fig. 14 shows the map in the (δ, β) parameter plane for γ = 7. Here a new domain
6 appears with peculiar properties. (More precisely, this domain appears already for
γ = 3 but in that case its size is very small and thus it is not seen in Fig. 13.) For
parameters from domain 6, the pattern of species spread exhibits nonuniqueness subject
to the initial conditions. Depending on u 0 , v0 , ∆u , ∆v , species invasion takes place either
through propagation of solitary pulses (Figs. 6 and 7) or through propagation of periodical
waves with irregular wave-makers (Fig. 5). Asterisk indicates the sub-domains where the
population wave of prey travels faster than the wave of predator. In the case of propagating
population pulses, it means that the pulse width grows with time. In the case of periodic
waves, it means that they actually propagate into the region already invaded by prey. One
can clearly see the tendency that an increase in γ makes prey ‘more invasive’ compared to
predator: the larger γ is, the higher the dashed curve lays.
In order to estimate threshold values of γ for which the bifurcations of the (δ, β) plane
take place, we accomplished an additional series of computer experiments. We obtained
that domain 2 of solitary pulses arises for γ ∗ ≈ 1.3. As for domain 6, we observe that it
arises at about the same value of γ ≈ 1.3; however, for γ < 5 its width is very small and
it can hardly be seen in the parameter plane.
The maps in the (δ, β) plane give important information about possible transitions
between different regimes that may occur as a result of system response to parameter
changes. For the sake of simplicity, we consider the situation when only δ can change
and all other parameters are fixed. Let us start with the case when δ is small. Since
δ is (dimensionless) predator mortality, small δ likely means that prey is under strong
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pressure from the predator. Thus, it is not surprising that small δ typically corresponds
to species extinction (unless γ is sufficiently large and β is sufficiently small, cf. the
left-hand bottom corner of Fig. 14). In order to avoid multivalence, we restrict further
analysis to the case γ = 3. An increase in δ makes this pressure smaller and extinction
changes, depending on other parameter values, either to local invasion (domain 5) or to
geographical spread through pulse propagation (domain 2). Note that, since the impact
of predation is still too strong, none of these two regimes lead to global persistence
of the invading species. However, further increase in δ changes these regimes first to
patchy invasion (either preceded by population front propagation, cf. Figs. 3 and 4, or not,
cf. Fig. 8) and then to invasion through propagation of smooth population fronts, see Fig. 2.
Similar analysis can be made varying other parameters. In particular, it is straightforward
to see that similar succession of regimes takes place when β is varied from a large value to
a small one.
5. Concluding remarks
The impact of predation on the spread of invasive species has long been an issue of
significant interest (Murray, 1989; Sherratt et al., 1995; Shigesada and Kawasaki, 1997;
Fagan and Bishop, 2000; Owen and Lewis, 2001; Petrovskii et al., in press). In most
cases, however, these studies were reduced to the case of populations with logistic growth.
Meanwhile, for many ecological populations their growth rate is believed to be damped
by the Allee effect (Allee, 1938; Berryman, 1981). Although it was shown that the Allee
effect can change population dynamics significantly (Dennis, 1989; Lewis and Kareiva,
1993; Courchamp et al., 1999; Gyllenberg et al., 1999; Petrovskii et al., 2002a), its impact
on species invasion has not been investigated in detail [but see Morozov (2003)]. In this
paper we have shown, using a predator–prey system as a paradigm, that the spatiotemporal
dynamics of invading species can become much more complex under the influence of the
Allee effect and exhibit regimes of invasion that have not been studied theoretically before.
A predator–prey system with the Allee effect for prey can exhibit such patterns of spread
as a patchy invasion (Fig. 8), geographical invasion without regional persistence (Figs. 6
and 7) and local invasion without geographical spread (Fig. 11). Remarkably, similar
patterns are often observed in nature. In contrast, a predator–prey system without Allee
effect only predicts species spread through either smooth population waves or travelling
fronts with population oscillations in the wake (similar to what is shown in Figs. 2, 3 and 4
respectively) [cf. Sherratt et al. (1995), Petrovskii et al. (1998), Petrovskii and Malchow
(2000)].
By means of extensive numerical simulations, we revealed the structure of the parameter
space of the system. That structure gives important information about possible transitions
between different regimes of invasion. Here we consider one example how it can enhance
our understanding of various aspects of biological invasion. It is well-known that, between
the stage of exotic species establishment in the new environment and the stage of its
geographical spread, there often exists a time-lag that can be sometimes as long as years,
or even decades. The nature of this time-lag is widely seen in species adaptation to the new
conditions. However, no specific dynamical mechanism has ever been proposed to explain
S. Petrovskii et al. / Bulletin of Mathematical Biology 67 (2005) 637–661
659
how species regional persistence actually changes to its geographical invasion. Our results
seem to suggest such a mechanism. In an introduced species, the individual fitness under
new environmental conditions is likely to be low which means that the species is more
prone to predation and to environmental stochasticity, so that the species is more likely to
go extinct when it is at low density. Virtually, it means large β and/or small δ. In terms of
the diagrams shown in Figs. 12–14, it means that it falls either to domain 1 corresponding to
species extinction or, in case fitness is not very low, to domain 5 corresponding to regional
persistence. As a result of species adaptation to the new environment, the individual fitness
is likely to increase so that the parameters move from domain 5 either to domain 3
or to domain 4; in both cases the regime of regional persistence gives way to species
geographical spread.
The results shown in Figs. 2–14 were obtained for = D2 /D1 = 1. However, we want
to emphasize that this not a principal limitation and the particular value = 1 was mainly
chosen in order to exclude another parameter. Our tentative numerical simulations made
for 0.5 < < 2 show that all the regimes described above exist also in that case, although
the position of the domains in the (δ, β) plane is somewhat different.
The initial conditions that we used in numerical simulations correspond to the problem
of biological control, see the lines after Eqs. (7) and (8). (Note that, in our computer
experiments, the radius of the domain initially inhabited by predator is always smaller
than that for prey.) Another biologically interesting case could be given by the situation
when either prey is introduced into an ecosystem where predator is established, or predator
is introduced into an ecosystem where prey is established. Mathematically, it means that
only one of the functions u(x, 0) and v(x, 0) is finite. Although this problem still remains
to be investigated in detail, we fulfilled some tentative simulations to make an early insight
into the corresponding system dynamics. Our results indicate that, in this case, the number
of possible invasion regimes is likely to be less than it was for the problem (5)–(8);
in particular we failed to observe the regimes shown in Figs. 6–11.
In this paper, our study of biological invasion has been restricted to the 1-D case. That
was done mainly for practical reasons because the time needed to fulfil necessary computer
simulations increases essentially in the case of two spatial dimensions. However, we want
to mention that, although a regular investigation of this issue is still lacking, our tentative
computer experiments indicate that the invasion patterns described above exist in the 2-D
predator–prey system as well [cf. also Petrovskii et al. (2002a,b), Morozov (2003)].
In conclusion, we want to mention that the structure of the (δ, β) plane tends to
become more complex with an increase in γ . This tendency can probably be easier
understood if placed into a more general context of self-organization in dynamical systems.
It is well-known from other branches of natural science that the dynamics of an open
system becomes the more complicated the ‘more open’ it is, i.e., the higher is the
energy/mass input into a given system (Prigogine, 1980; Haken, 1983). In particular, a
higher energy/mass input drives the system further from its equilibrium state and enhances
formation of complex spatiotemporal patterns. [A classical example of this increasing
complexity is given by the transition between laminar and turbulent flows in a pipe
which takes place when the speed of the fluid (and thus the mass/energy input) becomes
sufficiently high.] In the predator–prey model (1) and (2) the role of energy input that keeps
the system away from the trivial extinction equilibrium is played by the biomass increase
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due to prey population growth. The prey growth is quantified by parameter γ ; thus, a higher
value of γ makes the system ‘more open’ and increases its spatiotemporal complexity.
Acknowledgements
This work was partially supported by Russian Foundation for Basic Research under
grants 03-04-48018 and 04-04-49649, by U.S. National Science Foundation under grants
DEB-0080529 and DEB-0409984, by the University of California Agricultural Experiment
Station and by UCR Center for Conservation Biology.
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