J. Math. Biol. (2014) 68:1731–1756
DOI 10.1007/s00285-013-0687-1
Mathematical Biology
Analysis of the Trojan Y-Chromosome eradication
strategy for an invasive species
Xueying Wang · Jay R. Walton · Rana D. Parshad ·
Katie Storey · May Boggess
Received: 28 June 2012 / Revised: 3 May 2013 / Published online: 24 May 2013
© Springer-Verlag Berlin Heidelberg 2013
Abstract The Trojan Y-Chromosome (TYC) strategy, an autocidal genetic biocontrol
method, has been proposed to eliminate invasive alien species. In this work, we analyze the dynamical system model of the TYC strategy, with the aim of studying the
viability of the TYC eradication and control strategy of an invasive species. In particular, because the constant introduction of sex-reversed trojan females for all time
is not possible in practice, there arises the question: What happens if this injection is
stopped after some time? Can the invasive species recover? To answer that question,
we perform a rigorous bifurcation analysis and study the basin of attraction of the
recovery state and the extinction state in both the full model and a certain reduced
model. In particular, we find a theoretical condition for the eradication strategy to
work. Additionally, the consideration of an Allee effect and the possibility of a Turing
instability are also studied in this work. Our results show that: (1) with the inclusion
of an Allee effect, the number of the invasive females is not required to be very low
X. Wang (B) · J. R. Walton · M. Boggess
Department of Mathematics, Texas A & M University, College Station, TX 77843, USA
e-mail: [email protected]
J. R. Walton
e-mail: [email protected]
M. Boggess
e-mail: [email protected]
R. D. Parshad
Division of Mathematical and Computer Sciences and Engineering, King Abdullah University
of Science and Technology, 23955 Thuwal, Kingdom of Saudi Arabia
e-mail: [email protected]
K. Storey
Department of Mathematics, Carleton College, Northfield, MN 55057, USA
e-mail: [email protected]
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X. Wang et al.
when the introduction of the sex-reversed trojan females is stopped, and the remaining Trojan Y-Chromosome population is sufficient to induce extinction of the invasive
females; (2) incorporating diffusive spatial spread does not produce a Turing instability, which would have suggested that the TYC eradication strategy might be only
partially effective, leaving a patchy distribution of the invasive species.
Keywords Trojan Y-Chromosome eradication strategy · Extinction · Recovery ·
Allee effect · Turing instability
Mathematics Subject Classification (2000)
92D25 · 92D40 · 34A34
1 Introduction
Exotic species, commonly referred to as invasive species, are defined as any species,
capable of propagating themselves into a nonnative environment. Once established,
they can be extremely difficult to eradicate, or even manage (Hill and Cichra 2005;
Shafland and Foote 1979). Numerous cases of environmental harm and economic
losses are attributed to various invasive species (Harmful 1993; Myers et al. 2000).
Some well-known examples of these species include the burmese python in southern
regions of the United States, the cane toad in Australia, and the sea lamprey and round
goby in the Great Lakes region in the northern United States (Gutierrez 2005). The
cane toad was brought into Australia from Hawaii in 1935 in order to control the cane
beetle. These have multiplied rapidly since then, and are currently Australia’s worst
invasive species (Philips and Shine 2004). The sea lamprey entered the great lakes
in the 1800s, through shipping canals (Bryan et al. 2005), and these predators have
caused a severe decline in lake fish populations, adversely affecting local fisheries. The
burmese python population has been on the rise exponentially, in the Florida glades,
since the 1980s. Much of this is attributed to the exotic pet trade in Florida. These
invaders have had quite a negative impact on the local ecology, and climatic factors
might support their spread to a third of the United States (Rodda et al. 2005). Apart
from being considered one of the greatest threats to biodiversity, invasive species cause
agricultural losses worldwide in the range of several 100 billion dollars (Pimentel et al.
2005; Gutierrez 2005). Despite the magnitude of threats they pose, there have been no
conclusive results to eradicate or contain these species in real scenarios, in the wild.
A possible approach to the containment of these invasive species is the use of
a feasible biological control. It is well-known that the size of a population can be
seriously affected as a result of shifting the sex ratio (Hamilton 1976). The sterile insect
technique (SIT), works via the introduction of large quantities of sterile males, into
a target population, to compete with fertile males in mating (Knipling 1955). Mating
with sterile males does not produce any progeny, and the goal is to try and shift the sex
ratio so that there is an abundance of males, leading to less and less number of females
over time. The method is commonly used in Florida, to control crop damage via
Mediterranean fruit flies (Klassen and Curtis 2005). This in essence, is the underlying
principle for a number of recently proposed models (Howard et al. 2004; Gutierrez
and Teem 2006; Bax and Thresher 2009), that use genetic modification to control
invasive fish species. Among these models, the Daughterless Carp (Bax and Thresher
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Analysis of the TYC eradication strategy for an invasive species
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2009) and the Trojan Y-Chromosome (TYC) (Gutierrez and Teem 2006), have been
particularly well studied. The first strategy aims at causing extinction via the release of
individuals with chromosomal insertions of aromatase inhibitors. The latter strategy
tries to cause extinction via the constant release of sex-reversed supermales, denoted
as r in (2.1), into a target population, see Fig. 1. The idea behind the TYC strategy,
is that mating with the sex-reversed supermales results in only male or supermale
progeny, leading to less and less number of females over time, and ultimately driving
the female population to zero. This will thus lead to extinction of the population. Note
that the TYC strategy only works for a few species with X Y sex determination system,
that are capable of producing viable progeny during the sex reversal process. Thus,
it is not applicable to all the species mentioned earlier. The TYC eradication strategy
was proposed by Gutierrez and Teem (2006). Parshad and Gutierrez (2010) rigorously
proved the existence of a global attractor for the model derived in (Gutierrez and
Teem 2006), with the inclusion of diffusion. In particular, they show that the attractor
possesses the extinction state, if the introduction of sex-reversed supermales is large
enough. However, these works do not explore the full structure of the attractor, via a
rigorous stability and bifurcation analysis. Changing the sex ratio of a population seems
difficult to achieve in practice, there have been a number of studies to this end (Nagler
et al. 2001). It has also been reported that environmental pressure for masculinization
would lead to extinction of fish populations with a XY sex-determination system
(Hurley et al. 2004). Masculinization by exposure to androgens has been reported in
the wild (Katsiadakiand et al. 2002), and is common in the salmon industry to produce
an all-male stock (Hunter and Donaldson 1983).
In order to develop successful physical and biological controls for exotic fish
species, the dynamics of fish populations and eradication strategies constructed on
the basis of these dynamics have to be studied and further understood. The goal of this
work is to pursue a number of questions, that are not addressed in (Gutierrez and Teem
2006; Parshad and Gutierrez 2010). These earlier works in essence, show that extinction is possible if the rate of release μ, of sex-reversed supermales is large enough.
They do not address the case of small μ. We now ask the following question: what
happens if the introduction of the feminized supermales is stopped? Can the fish population possibly recover, from low levels, instead of going to the extinction state? Any
eradication strategy, with potential to be implemented into a management practice,
cannot possibly sustain the constant introduction of genetically modified organisms
into a target population, particularly over a large spatial domain. Thus, a rigorous stability analysis of the model, in order to understand the specific structure of the attractor
has to be performed, in the case this release rate μ > 0, and μ = 0. In particular, the
extinction or recovery, and their connection with the range of μ, has to be rigorously
identified. This is the first contribution of the current work. Next, the Allee effect (a
well-known phenomenon from population dynamics, defined as a positive correlation
between population density and individual fitness) (Odum and Allee 1954; Stephens
and Sutherlan 1999) is incorporated into the current TYC model. Turing instability is
also explored in the context of the TYC model.
This article is organized as follows. Section 2 introduces the TYC eradication
strategy and the TYC model developed in (Gutierrez and Teem 2006). Section 3
presents the equilibrium and stability analysis of the TYC model. The Allee effect is
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X. Wang et al.
studied in Sect. 4. The basin of attraction of the extinction state is demonstrated in
Sect. 5 in terms of a reduced two-dimensional TYC model, the full TYC model when
μ = 0, and a reduced two-dimensional TYC model with the inclusion of an Allee
effect. In Sect. 6, we show that incorporating diffusive spatial spread does not give
rise to a Turing instability. Concluding remarks are given in Sect. 7.
2 Model description
Gutierrez and Teem (2006) proposed a mathematical model of the Trojan YChromosome (TYC) eradication strategy for invasive fish species. The idea behind
this strategy is to eliminate the wild-type female fish from a population, by introducing a sex-reversed “Trojan” female fish, into the population. Mating with the trojan,
only leads to male, or supermale progeny. This will lead to lesser and lesser wild-type
females over time, ultimately leading to their eradication, and thus causing the population to go extinct. The strategy relies on two facts: (1) the wild-type invasive fish
species are required to have an XY sex-determination system, that is, sex is determined
by the presence of a Y chromosome in a fish species; (2) the sex change of genetic
males to females and of females to males can be made through chemical induced
genetic manipulation, such as hormone treatments.
Gutierrez and Teem (2006) considered the Nile tilapia as an example. Their TYC
model describes the dynamics of four phenotype/genotype variants of the invasive
fish species: genotype XX females, genotype XY males, phenotype YY supermales
and phenotype sex-reversed YY females, denoted as f, m, s and r , respectively. The
eradication strategy works as follows. A sex-reversed trojan female (r ), carrying two
Y Chromosomes, is introduced into a wild-type invasive fish population comprised
of wild-type males (m) and females ( f ). Mating between a phenotype female (r )
and a genotype male (m) produces a disproportionate number of male fish m. The
high incidence of males decreases the sex ratio of females to males. Ultimately, the
wild-type females would be eliminated from the population. As a consequence, the
wild-type invasive population goes extinct.
It follows from the pedigree tree of the TYC model illustrated in Fig. 1, that the
equation describing the time evolution of the system can be written as:
Fig. 1 The pedigree tree of the TYC model (that demonstrates Trojan Y-chromosome eradication strategy).
a Mating of a wild-type XX female (f) and a wild-type XY male (m). b Mating of a wild-type XY male
(m) and a sex-reversed YY female (r). c Mating of a wild-type XX female (f) and a YY supermale (s).
d Mating of a sex-reversed YY trojan female (r) and a YY supermale (s). Red color represents wild types,
and white color represents phenotypes (color figure online)
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Analysis of the TYC eradication strategy for an invasive species
df
dt
dm
dt
ds
dt
dr
dt
1735
1
f mβ L − δ f,
2
1
1
=
f m + r m + f s β L − δm,
2
2
1
=
r m + r s β L − δs,
2
=
= μ − δr,
(2.1)
where f, m, s, r define the number of individuals in each associated class; positive
constants β and δ represent the per capita birth and death rates, respectively; nonnegative constant μ denotes the rate at which the sex-reversed YY females r are introduced;
L is a logistic term given by
L =1−
f +m +s +r
,
K
(2.2)
with K interpreted as the carrying capacity of the ecosystem.
3 Equilibrium analysis of the TYC model
Let ( f ∗ , m ∗ , s ∗ , r ∗ )T denote an equilibrium of model (2.1). We are interested in the
solution in the region Ω := {( f, m, s, r )T : 0 ≤ f, m, s, r ≤ K }.
3.1 The case μ > 0
Assume that the sex-reversed trojan females are introduced into the wild at the rate
μ > 0. We study the dynamics of the associated TYC model.
3.1.1 Determination of equilibria
Case 1 Assume f ∗ = 0.
An equilibrium with f ∗ = 0 describes a steady state of (2.1) at which wild-type
XX females are eradicated. Let
Γ = βμ2 − β K δμ + K δ 3 .
(3.1)
If Γ ≥ 0, Eq. (2.1) has only one equilibrium that satisfies f ∗ = 0, namely,
μ T
. Otherwise, if Γ < 0, Eq. (2.1) has two equilibria with f ∗ = 0.
0, 0, 0,
δ
δ 2 μ μ T
μ T
− ,
, and the other one is 0, 0, K 1 −
. Clearly, if
One is 0, 0, 0,
δ
μβ
δ δ
f ∗ > 0, m ∗ > 0 and r ∗ > 0, then s ∗ > 0, and s ∗ = 0 is not a solution of biological
meaning.
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X. Wang et al.
Case 2 Assume f ∗ > 0.
By direct computations, we find that the equilibrium satisfies
(r ∗ )2 (r ∗ + s ∗ )
,
s ∗ (s ∗ − r ∗ )
2r ∗ s ∗
m∗ = ∗
,
s − r∗
μ
r∗ = ,
δ
f∗ =
(3.2)
where s ∗ satisfies
s(as 3 + bs 2 + cs + d) = 0,
(3.3)
with a = a(r ∗ ) := r ∗ β, b = b(r ∗ ) := 2β(r ∗ )2 − β K r ∗ + δ K , c = c(r ∗ ) :=
β K (r ∗ )2 − 2δ K r ∗ , and d = d(r ∗ ) := β(r ∗ )4 + K δ(r ∗ )2 .
If μ > 0, assumptions of model (2.1) on the parameter values yield a > 0 and
d > 0. Clearly, s ∗ > r ∗ > 0 is required for the equilibrium of model (2.1) to be in the
region Ω.
K 2 δK
K
2δ
±
±
and rc = . It is easy to verify that rb± and rc are
−
Let rb =
4
4
2β
β
the nonzero roots of b(r ∗ ) = 0 and c(r ∗ ) = 0, respectively.
We define α = β K /2δ.
Lemma 3.1 (a) If α = 4, rb− = rc = rb+ , (b) If α > 4, then rb− < rc < rb+ .
Proof One can easily verify the result by direct calculation.
For simplicity, we write Δ = 18abcd −
+
−
−
In fact, Δ
is the determinant of a cubic equation as 3 + bs 2 + cs + d = 0. The three cases for
this determinant are well known.
4b3 d
(bc)2
4ac3
27(ad)2 .
Proposition 3.1 Assume that Δ > 0.
μ
(a) If α > 4, then Eq. (3.3) has two positive roots for < rb+ , and no positive roots
δ
μ
for ≥ rb+ .
δ
μ
< rc , and no positive roots
(b) If α ≤ 4, then Eq. (3.3) has two positive roots for
δ
μ
for ≥ rc .
δ
Proof We define the term in parentheses of Eq. (3.3) to be
g(s) = as 3 + bs 2 + cs + d.
(3.4)
By a > 0 and Δ > 0, Eq. (3.4) has three distinct real roots. Moreover, none of the
roots can be zero because d = 0.
Case (a) α > 4. By Lemma 3.1, we have rb− < rc < rb+ .
(1) Assume r ∗ = μ/δ ∈ (0, rb− ). Then b > 0 and c < 0. The signs of the coefficients
of g(s) and g(−s) are
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Analysis of the TYC eradication strategy for an invasive species
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sign(a, b, c, d) = (+, +, −, +),
(3.5)
sign(−a, b, −c, d) = (−, +, +, +),
(3.6)
and
respectively. So there are 2 sign changes in g(s) and 1 sign change in g(−s). By
Descartes’ rule of signs, we see that g(s) = 0 has either 2 or 0 positive roots and
exactly 1 negative root. Since g(s) = 0 has three nonzero real roots, the roots
have to be comprised of 2 positive ones and 1 negative one.
(2) Assume r ∗ ∈ [rb− , rb+ ). If r ∗ = rb− , then b = 0 and c < 0. If r ∗ ∈ (rb− , rc ), then
b < 0 and c < 0. If r ∗ = rc , then b < 0 and c = 0. If r ∗ ∈ (rc , rb+ ). then b < 0
and c > 0. By similar arguments, we can show that g(s) has two positive zeros.
(3) Assume r ∗ ∈ [rb+ , ∞). If r ∗ = rb+ , then b = 0 and c > 0. If r ∗ > rb+ , then b > 0
and c > 0. Since there are no changes of sign in the coefficients, there are no
positive zeros for g(s).
Case (b) α ≤ 4. Then b ≥ 0. Similarly, we can show that:
(1) if r ∗ < rc , then c < 0 and g(s) has two positive zeros;
(2) if r ∗ ≥ rc , then c ≥ 0 and g(s) has no positive zeros.
The results follow, since all nonzero roots of (3.3) satisfy g(s) = 0.
Proposition 3.2 Assume that Δ = 0.
μ
(a) If α > 4, then Eq. (3.3) has either zero or two positive roots for < rb+ , and no
δ
μ
positive roots for ≥ rb+ .
δ
μ
(b) If α ≤ 4, then Eq. (3.3) has either zero or two positive roots for < rc , and no
δ
μ
positive roots for ≥ rc .
δ
Proof If Δ = 0, Eq. (3.4) has a multiple root and all its roots are real. The results
follow from Descartes’ rule of signs.
Proposition 3.3 Assume that Δ < 0. Then Eq. (3.3) has no positive roots.
Proof If Δ < 0, g(s) = 0 has one real root and two complex conjugate roots. The
results can be shown by Descartes’ rule of signs.
3.1.2 Stability of equilibria
Linearize system (2.1) at the equilibria. The associated Jacobian matrix is given by
⎡β
β ∗ ∗
m∗ L ∗ − ξ − δ
f L −ξ
⎢ 2
2
⎢ (m ∗ /2 + s ∗ )β L ∗ − η ( f ∗ /2 + r ∗ /2)β L ∗ − η − δ
Jq = ⎢
⎣ −κ
βr ∗ L ∗ /2 − κ
0
0
−ξ
β f ∗ L∗ − η
βr ∗ L ∗ − κ − δ
0
−ξ
⎤
⎥
⎥
m ∗ β L ∗ /2 − η
⎥
∗
∗
∗
(m /2 + s )β L − κ ⎦
−δ
(3.7)
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X. Wang et al.
where L ∗ = 1 − ( f ∗ + m ∗ +s ∗ +r ∗ )/K, ξ = β f ∗ m ∗ /(2K ), η = β( f ∗ m ∗ +r ∗ m ∗ +
2 f ∗ s ∗ )/(2K ) and κ = βr ∗ m ∗ + 2s ∗ /(2K ).
The equilibrium of model (2.1) that satisfies f ∗ = 0 is of the form of (0, 0, s ∗ , r ∗ )T .
Evaluate (3.7) at (0, 0, s ∗ , r ∗ )T . We find the associated characteristic equation can be
written as:
β ∗ ∗
s∗
r L −δ
λ − βr ∗ L ∗ −
−δ (λ + δ) = 0, (3.8)
(λ + δ) λ −
2
2K
and the corresponding eigenvalues are given by
λ1 = −δ
β
λ2 = r ∗ L ∗ − δ
2
s∗ λ3 = βr ∗ L ∗ −
−δ
K
λ4 = −δ.
(3.9)
Hence, the eigenvalues associated with p := (0, 0, 0, μ/δ)T are λ1 = −δ, λ2 =
− δ = −Γ /(2δ 2 K ) − δ/2, λ3 = βr ∗ L ∗ − δ = −Γ /(δ 2 K ) and λ4 = −δ.
Clearly, λ1 , λ4 < 0. Thus, the stability of p will be determined by λ2 and λ3 . If Γ ≥ 0,
then λ2 < λ3 ≤ 0 and p is locally stable. If Γ < 0, then λ3 > 0, and p is unstable.
Recall that if Γ < 0, the system (2.1) has two equilibria: p and q := 0, 0, K 1 −
δ 2 μ μ T
− ,
. One can easily verify that the eigenvalues associated with q are
μβ
δ δ
λ1 = −δ, λ2 = −δ/2, λ3 = −βr ∗ s ∗ /K and λ4 = −δ. These eigenvalues are all
negative, and hence q is locally stable.
So, if Γ ≥ 0, p represents the elimination state (ES) of the system; otherwise,
Γ < 0 and q represents the ES.
βr ∗ L ∗ /2
3.1.3 A condition for the eradication strategy
Let
(H 1) : ≥ 0, α > 4, and μ/δ ≥ rb+ ;
(3.10)
(H 2) : ≥ 0, α ≤ 4, and μ/δ ≥ rc ;
(H 3) : < 0.
(3.11)
(3.12)
By Propositions 3.1 and 3.3, if one of three hypothesis (H1)–(H3) is valid, the TYC
model will only have the equilibria that satisfy f ∗ = 0 when μ > 0. In other words,
(H1)–(H3) provides a sufficient condition for the eradication strategy to work. Altogether with the analysis from Sect. 3.1.2, we have the following result.
Theorem 3.1 Suppose μ > 0. If one of three hypothesis (H1)–(H3) holds, the eradication strategy works. Moreover, if Γ < 0, the TYC model (2.1) has two equilibria:
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Analysis of the TYC eradication strategy for an invasive species
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δ 2 μ μ T
− ,
p := (0, 0, 0, μ/δ)T and q := 0, 0, K 1−
, and p is locally unstable
μβ
δ δ
whereas q is locally stable; if Γ ≥ 0, then the system (2.1) has only one equilibrium p
and it is locally stable. In either case, the stable equilibrium represents the elimination
state of the wild-type species.
3.2 The case μ = 0
3.2.1 Equilibrium analysis
According to Theorem 3.1, the continual injection of the sex-reversed supermales can
eliminate the wild-type females from the population. However, it is not practical to
keep the introduction constant for all time. So, a natural question is: what happens if
the introduction of the trojan fish is stopped after some time, that is, μ = 0?
If μ = 0, we can verify by direct computation that the equilibrium of model (2.1)
is of the form
( f ∗ , m ∗ , 0, 0)T , with f ∗ = m ∗ .
(3.13)
For simplicity, we write φ = 16δ/β K . Clearly, φ > 0 since β, K and δ are all assumed
√
K
to be positive. Let f ±∗ = (1 ± 1 − φ). If φ < 1, the TYC model (2.1) has three
4 ∗
distinct equilibria for which f = f +∗ , f −∗ and 0, respectively. If φ = 1, the model
K
and 0, respectively. Otherwise if φ > 1, the system
has two equilibria and f ∗ =
4∗
has only one equilibrium and f = 0.
Evaluate the Jacobian matrix (3.7) at the equilibria. In the case φ ≤ 1, we find that
the corresponding characteristic equation is given by
3 λ + δ λ − (2γ − δ) = 0,
(3.14)
3f∗
β where γ = 0, or γ = γ± := f ±∗ 1 − ± . The associated eigenvalues are λi =
2
K
−δ < 0, for i = 1, 2, 3, and λ4 = −δ, 2γ− − δ or 2γ+ − δ. Therefore, the sign of λ4
determines the local stability of the equilibrium. In particular, at (0, 0, 0, 0)T , γ = 0
and hence λi = −δ < 0 for all 1 ≤ i ≤ 4.
In addition, it follows from direct calculation that
2γ± − δ = −
2δ 1−φ± 1−φ .
φ
(3.15)
Then φ ≤ 1 yields 2γ+ − δ < 0 and 2γ− − δ > 0. It demonstrates that the equilibrium (0, 0, 0, 0)T (which represents the elimination state) and ( f +∗ , f +∗ , 0, 0)T are
locally stable (which represents the recovery state), whereas ( f −∗ , f −∗ , 0, 0)T is locally
unstable, and is a saddle.
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X. Wang et al.
4 The Allee effect
The (component) Allee effect is a positive relationship between any measurable component of individual fitness and population size (Odum and Allee 1954; Stephens and
Sutherlan 1999). Population-dynamic consequences of the Allee effect are of primary
importance in conservation biology and other fields of ecology.
Due to the difficulty of finding mates at low population sizes, we introduce the
Allee effect into the mathematical model of the TYC eradication strategy. A reasonable
model (2.1) with the inclusion of an Allee effect takes the form:
df
dt
dm
dt
ds
dt
dr
dt
f
1
fm
− 1 β L − δ f,
2
f0
1
f
r
1
f
=
fm
− 1 + rm
−1 + fs
− 1 β L − δm,
2
f0
2
r0
f0
1
r
=
rm + rs
− 1 β L − δs,
2
r0
=
= μ − δr,
L = 1−
f +m +s +r
,
K
(4.1)
where f 0 (resp. r0 ) is a critical value of f fish (resp. r fish), which is used to characterize
the minimal group size to find mates, rear offspring, search for food and/or fend off
attacks from predators. Mathematically, we assume that f 0 K , and r0 = f 0 .
We denote this model (4.1) by ATYC model. Here our use of the cubic form for
incorporating an Allee effect is for illustrative purposes only in the same spirit as its
use in (Lewis and Kareiva 1993), and in the classical KPP equation (Kolmogorov et
al. 1937).
The eradication strategy works in the way that the wild-type female population can
be eliminated by introducing the trojan fish. We have seen from the results in Sect.
3.2 that the wild-type fish will either go extinct or recover if the injection of the trojan
fish is removed. Now the question is when would be the optimal termination time for
such a removal? Here, by the “optimal termination time”, we mean the minimal time
that is required to switch off the injection of the trojan fish so that the wild-type fish
species won’t be able to recover thereafter. Let τ0 be the optimal termination time.
Assume μ = μ0 if t ≤ t0 and μ = 0 otherwise. The top panel in Fig. 2 illustrates that,
in the original TYC model (2.1), the introduction of the trojan fish has to last until the
number of the wild-type females, f , is extremely low to guarantee that the wild-type
fish will not recover. In this case, τ0 ≈ 58 (time unit), and f (τ0 ) < 1, which indeed
provides no information for biological implementation. The middle (resp. the bottom)
panel in Fig. 2 displays the associated results for the ATYC models with f 0 = 0.01K
(resp. f 0 = 0.02K ). Specifically, if f 0 = 0.01K , τ0 ≈ 34 and f (τ0 ) ≈ 21 (which
is about 0.07K ); if f 0 = 0.02K , τ0 ≈ 31 and f (τ0 ) ≈ 30 (which is about 0.1K ).
Therefore, it indicates that: (1) the number of the wild-type female fish at the optimal
termination time τ0 needs not be very low for the ATYC model; (2) the remaining
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Analysis of the TYC eradication strategy for an invasive species
300
XX female (f)
XY male (m)
YY super male (s)
YY female (r)
250
300
XX female (f)
XY male (m)
YY super male (s)
YY female (r)
250
200
200
150
150
100
100
50
50
0
1741
0
0
50
100
150
200
250
300
0
50
100
time
150
200
250
300
time
(a)
(b)
300
XX female (f)
XY male (m)
YY super male (s)
YY female (r)
250
300
250
200
200
150
150
100
100
50
50
0
XX female (f)
XY male (m)
YY super male (s)
YY female (r)
0
0
50
100
150
200
250
300
0
50
100
time
150
200
250
300
time
(c)
(d)
300
XX female (f)
XY male (m)
YY super male (s)
YY female (r)
250
300
250
200
200
150
150
100
100
50
50
0
XX female (f)
XY male (m)
YY super male (s)
YY female (r)
0
0
50
100
150
200
250
300
0
time
(e)
50
100
150
200
250
300
time
(f)
Fig. 2 The Allee effect is illustrated by comparing the solutions of the TYC model and those of ATYC
model I. The top panel (resp. the middle and the bottom panels) shows the solutions of TYC model (2.1)
(resp. the ATYC model (4.1) with f 0 = 3 and f 0 = 6) at different termination time of t0 . Here μ = 20 if
t ≤ t0 , and μ = 0 if t > t0 . β = 0.1, δ = 0.1 and K = 300 (color figure online)
123
1742
X. Wang et al.
trojan fish are sufficient to eliminate the wild-type females from the population after
τ0 .
5 The basin of attraction of the extinction state of a wild-type species when
µ=0
5.1 The reduced two-dimensional TYC model
The equilibrium analysis in Sect. 3.2.1 shows that both r and s would exponentially
decay to 0 if μ = 0. If r = s = 0 in the original model, (2.1) gives a reduced
two-dimensional system as follows:
1
f +m
df
= f mβ 1 −
− δ f,
dt
2
K
dm
1
f +m
= f mβ 1 −
− δm.
dt
2
K
(5.1)
Nondimensionalize model (5.1). Let x1 = f /K , x2 = m/K and τ = δt. Recall that
α = β K /(2δ). We get
d x1
= αx1 x2 (1 − x1 − x2 ) − x1 ,
dτ
d x2
= αx1 x2 (1 − x1 − x2 ) − x2 .
dτ
(5.2)
It can be seen from the bifurcation diagram for model (5.2) which is displayed
in Fig. 3, if α < 8 (i.e., φ = 16δ/(β K ) = 8/α > 1), there is only one stable
equilibrium. If α = 8 (i.e., φ = 1), there are two equilibria: one is stable and the other
one is unstable. If α > 8 (i.e., φ < 1), there are three equilibria: one is unstable and the
other two are stable. Focus on the most interesting case
√where α > 8. One
√ can verify
=
(0,
0),
p
=
((1
−
1
−
φ)/4,
(1
−
1 − φ)/4),
that the three equilibria
are:
p
0
u
√
√
and ps = ((1 + 1 − φ)/4, (1 + 1 − φ)/4). In particular, p0 represents the ES,
and ps represents the recovery state (RS). Let (x1∗ , x2∗ ) denote the equilibrium of (5.2).
0.4
0.3
x1
Fig. 3 Bifurcation diagram for
the reduced two-dimensional
model (5.2). The stable
equilibrium and the unstable
equilibrium are displayed by the
solid curve and the dash-dot
curve, respectively
0.2
0.1
0
0
123
5
10
α
15
20
25
Analysis of the TYC eradication strategy for an invasive species
1743
Write η = αx1∗ (1−3x1∗ ). The eigenvalues of the system (5.2) are given by λ1 = 2η −1
and λ2 = −1. One can verify that p0 and ps are locally stable, whereas pu is a saddle.
To study the basin of attraction for the ES, we analyze the dynamics of (5.2) at the
saddle pu . Let
cos(θ )
Rθ =
sin(θ )
− sin(θ )
cos(θ )
denote the rotation matrix. Consider the coordinate transformation
U
Δx1
= Rπ/4
,
Δx2
S
(5.3)
(5.4)
for which Δx1 = x1 − x1∗ and Δx2 = x2 − x2∗ . This yields
d
dt
U
S
=
λ1 0
0 λ2
U
S
+
F(U, S)
,
0
(5.5)
where
√
√
F(U, S) = − 2αx ∗ (U 2 + S 2 ) + α(1 − 4x ∗ )(U 2 − S 2 )/ 2 − α(U 2 − S 2 )U.
(5.6)
We now look for the curve on the (U, S) phase plane such that it forms the separatrix
between the basis of attraction for the ES and that for the RS. The separatrix indeed
is the stable manifold of the origin (U, S)T = (0, 0)T . Suppose such a curve exists.
Then it has to be of the form U = U (S) = Sh(S) with h(0) = 0, because U (0) =
U (0) = 0. By the invariant property on this curve, (5.5) yields
λ2 S(h(S) + Sh (S)) = λ1 Sh(S) + F(Sh(S), S).
(5.7)
In view of Eq. (5.7) and h(0) = 0, we have
1 −n
S
h(S) =
λ2
S
ξ n−2 F(ξ h(ξ ), ξ )dξ,
(5.8)
0
where n = 1 − λ1 /λ2 . It is clear that n > 1 because λ1 > 0 > λ2 . Substitute
λ1 = 2η − 1 and λ2 = −1 into the expression of n. We get n = 2η. It then follows
from the definition of F(U, S) in (5.6) that
F(U, S) = c20 U 2 + c02 S 2 + c30 U 3 + c12 U S 2 ,
(5.9)
√
√
for which c20 = α(1 − 6x1∗ )/ 2, c02 = α(−1 + 2x1∗ )/ 2, c30 = −α and c12 = α.
Substitute (5.9) and λ2 = −1 into (5.8). We have
123
1744
X. Wang et al.
1
1
x 1’(t) = 0
x 2 ’(t) = 0
0.8
x 1 ’(t) = 0
x 2 ’(t) = 0
0.8
W s (pu )
W s (pu )
0.6
x2
x2
0.6
0.4
pu
0.2
ps
0.4
ps
0.2
pu
0
p0
0
0
0.2
0.4
0.6
0.8
1
p0
0
0.2
0.4
x1
(a)
0.6
0.8
1
x1
(b)
Fig. 4 The separatrix W s (pu ) of the extinction and the recovery of the invasive fish species f , which
are displayed in bold solid curves. The dashed (resp. dotted) curve is the nullcline for x1 (t) = 0 (resp.
x2 (t) = 0). The solid and dashed curves are trial solutions of (5.2). The dashed ones converge to the
recovery equilibrium ps , whereas the solid ones converge to the extinction equilibrium p0
1
h(S) = −S
θ n c20 (h(Sθ ))2 + c02 + c30 Sθ (h(Sθ ))3 + c12 Sθ h(Sθ ) dθ.
0
(5.10)
In general, (5.10) has no closed-form analytical solution. We have to rely on the
numerical method to evaluate it (Atkinson 1992, 1997). Let W s (pu ) denote the stable
manifold at pu . Figure 4 shows W s (pu ) with different values of the parameter α. If
the initial position (x1 (0), x2 (0))T is below W s (pu ), the solution will converges to
p0 (which is illustrated by the directed solid curves in Fig. 4). This demonstrates the
extinction of the invasive species. Otherwise, if the initial (x1 (0), x2 (0))T is above
W s (pu ), the solution will converge to ps (which is illustrated by the directed dashed
curves in Fig. 4). This shows the recovery of the invasive species. Therefore, W s (pu )
serves as a separatrix that separates the extinction from the recovery. Besides, if α
increases, Fig. 4 shows that the separatrix W s (pu ) is moving towards an L-shaped
curve which comprises two segments of axes in the first quadrant restricted to the unit
square [0, 1] × [0, 1]. Thus, the results show that: (1) the basin of attraction of the ES
expands as α goes down to the bifurcation point 8; (2) the basin of attraction of the ES
dramatically decreases as α goes up and becomes unbounded. Moreover, if α 8, the
wild-type invasive fish can always be established, as along as their initial population
is not very low.
5.2 The full TYC model (2.1) when μ = 0
Returning to the full model (2.1), it is natural to ask whether the wild-type invasive
fish species will die off or recover after the removal of the introduction of sex-reversed
supermales, that is, μ = 0.
Let x1 = f /K , x2 = m/K , x3 = s/K and x4 = r/K . Rescale time τ = δt.
Nondimensionalize the TYC model (2.1) with μ = 0. We get
123
Analysis of the TYC eradication strategy for an invasive species
d x1
dτ
d x2
dτ
d x3
dτ
d x4
dτ
1745
= αx1 x2 L̂ − x1 ,
= α(x1 x2 + x2 x4 + 2x1 x3 ) L̂ − x2 ,
= α(x2 x4 + 2x3 x4 ) L̂ − x3 ,
= −x4 ,
(5.11)
4
where L̂ = 1 − i=1
xi . Define X = (x1 , x2 , x3 , x4 )4 and F to be vector of the right
hand sides of Eq. (5.11). Then Eq. (5.11) can be written as:
dX
= F(X).
dτ
(5.12)
In view of the results in Sect. 3.2, the equilibrium of (5.11) is of the form p ∗ =
(x ∗ , x ∗ , 0, 0)T , and the associated Jacobian matrix is given by
⎡
η−1
⎢
η
M = DF( p ∗ ) = ⎢
⎣0
0
η
η−1
0
0
−B
2A − B
−1
0
⎤
−B
η ⎥
⎥,
A ⎦
−1
(5.13)
where η = A − B with A = αx ∗ L̂ ∗ , B = α (x ∗ )2 and L̂ ∗ = 1 − 2x ∗ . If x ∗ = 0, then
M = −I4 where I4 represents the 4 × 4 identity matrix. If x ∗ = 0, then η = A − B =
αx ∗ (1 − 3x ∗ ) > 0. Let
⎡
⎤
T12 T13 −T23
T11
⎢ −T11 T22 T23 −T23 ⎥
⎥,
T=⎢
⎣0
⎦
T32 T33 0
0
0
1
0
where T11 = −A2 , T12 = (2 AB − 3A2 )/(2η), T13 = (−7A2 + 8AB − 2B 2 +
2ηB)/(4η2 ), T22 = A2 /(2η), T23 = A2 /(4η2 ), T33 = (2 A − B − η)/(2η), and
T32 = A. Then by the coordinate transformation X = p + TY, Eq. (5.12) yields
dY
= JY + G(Y),
dt
where J is the Jordan canonical form of M with
⎡
−1 1
0
⎢0
−1
1
J=⎢
⎣0
0
−1
0
0
0
(5.14)
⎤
0
⎥
0
⎥,
⎦
0
2η − 1
and G(Y) = T−1 (F( p + TY) − MTY).
123
1746
X. Wang et al.
Consider 0 < φ < 1. At pu , the eigenvalue σ4 := 2η − 1 > 0. Hence the Jordan
form (5.13) indicates that pu is hyperbolic of saddle type. Thus, there exists a solution
flow of equation (5.14), (t; ξ1 , ξ2 , ξ3 ) ∈ C 1 ([0, ∞), U ), for some neighborhood
U ⊂ R3 , and limt→∞ (t; ξ1 , ξ2 , ξ3 ) = 0. Specifically, for any C = (c1 , c2 , c3 , 0)T
with (c1 , c2 , c3 )T ∈ U, (t, ·) is given by
τ
(τ ; C) = (τ )C +
∞
(τ − s)G((s))ds −
s
u (τ − s)G((s))ds,
s
τ
0
where
⎡
e−τ
⎢
0
s (τ ) = ⎢
⎣0
0
τ e−τ
e−τ
0
0
1 2 −τ
2τ e
τ e−τ
e−τ
0
⎤
0
0⎥
⎥,
0⎦
0
and
⎡
0
⎢
0
u (τ ) = ⎢
⎣ .eps0
0
0
0
0
0
0
0
0
0
⎤
0
0 ⎥
⎥.
0 ⎦
eσ4 τ
Then system (5.14) processes a 3-dimensional local stable manifold at the origin 0,
s
Wloc
(0) = {(ξ1 , ξ2 , ξ3 , ξ4 )T : ξ4 = 4 (0; ξ1 , ξ2 , ξ3 ), (ξ1 , ξ2 , ξ3 )T ∈ U },
(5.15)
for which = (1 , 2 , 3 , 4 )T .
Let n = (n 1 , n 2 , n 3 , n 4 )T with n 1 = n 2 = 1, n 3 = −(T12 + T22 )/T32 and
n 4 = T33 (T12 + T22 ) − T32 (T13 + T23 ) /T32 , which denotes the normal vector of
the local stable manifold at pu . Let t0 denote the time at which the introduction of the
sex-reversed supermales r is switched off. Since pu is hyperbolic of saddle type, there
s (0) is well-defined and the following results
exists a neighborhood U0 such that Wloc
hold: (1) if X(t0 ) satisfies T−1 (X(t0 ) − pu ) ∈ U0 and T−1 (X(t0 ) − pu ) · n > 0, the
solution of Eq. (5.11) will converge to ps , which implies that the invasive fish species
f and m will recover after the removal of r (see the red curves in Fig. 5); (2) whereas
if T−1 (X(t0 ) − pu ) ∈ U0 and T−1 (X(0) − pu ) · n < 0, the solution will approach p0 ,
which indicates that f and m will go extinct after the trojan introduction is stopped
(see the blue curves in Fig. 5).
Therefore, by (5.11), whether the wild-type species will die off after the injection of
trojan females is stopped depends on the value of the parameter α and the initial state
of the trajectories. Specifically, if α is a constant, the stable manifold of pu provides
a separatrix of extinction and recovery. If the initial state is identical, increasing α
will result in the basin of the attraction of the extinction shrinking whereas that of the
recovery expanding (see Fig. 5).
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Analysis of the TYC eradication strategy for an invasive species
1747
Fig. 5 Projection of some trajectories of the TYC model (5.11) onto the x1 x2 plane. The red (resp. blue)
curves represents recovery (resp. extinction) of the wild-type invasive species. The associated initials of
these trajectories are in a close neighborhood of the saddle fixed point pu . The black curves show the
solutions of (5.11) that are very close to the projection of the separatrix of recovery and extinction onto the
x1 x2 plane. Here x3 = 1/300 and x4 = 10/300 (color figure online)
5.3 The reduced 2-dimensional ATYC model
We consider the case when r = s = 0 in the full ATYC model (4.1). It yields a reduced
two-dimensional system
1
df
= f mβ L
dt
2
dm
1
= f mβ L
dt
2
f
− 1 − δ f,
f0
f
− 1 − δm.
f0
(5.16)
Let x1 = f /K , x2 = m/K , x0 = f 0 /K and τ = δt. Nondimensionalizing model
(5.16) gives
d x1
= αx1 x2 (1 − x1 − x2 )(x1 /x0 − 1) − x1 ,
dτ
d x2
= αx1 x2 (1 − x1 − x2 )(x1 /x0 − 1) − x2 .
dτ
(5.17)
Bifurcation diagrams for the reduced model (5.17) are displayed in Fig. 6. Figure
6a shows the steady-state behavior of x1 (which is the dimensionless counterpart of f )
as a function of α when x0 is 0.05. Note that the stability of the system (5.17) changes
at the limit point, L P. There is a unique stable equilibrium point p0 = (0, 0)T if
α < L P (α ≈ 1.6 as x0 = 0.05). At α = L P, there are two equilibria and only the
lower branch is stable. If α > L P, there will be three equilibria: p0 (the lower branch),
pu (the middle branch) and ps (the top branch), for which pu is unstable and indeed a
saddle, and both p0 (representing the ES) and ps (repenting the RS) are stable. Thus, a
saddle-node bifurcation occurs at L P. If x0 varies, Fig. 6 b displays a two-parameter
bifurcation, which illustrates how α changes as x0 varies at the LP. In particular, it
123
1748
X. Wang et al.
0.4
0.5
0.3
0.4
x0
LP
x1
0.3
0.2
0.2
0.1
0.1
0
0
5
10
15
α
20
0
25
0
(a)
20
40
α
60
80
100
(b)
Fig. 6 Bifurcation diagram for the reduced two-dimensional model (5.17). a x1 as a function of α with
x0 = 0.05. The stable equilibrium (resp. unstable equilibrium) is displayed by the solid curve (resp. the
dash-dot curve). b Two-parameter bifurcation showing α as a unction of x0 at the limit point (LP)
1
0
x 1 ’(t) = 0
x 2 ’(t) = 0
x 1 ’(t) = 0
0.8
x 2 ’(t) = 0
0
0.8
0
1
w s (pu )
w s (pu )
0
0
00
u
ps
0
0.4
0
0
p
0
0
0.4
x2
ps
0
x2
0.6
0
0.6
pu
0.2
0
0.2
0
0
0
0
0
0
0
0.2
0.4
0.6
0.8
1
p
0
0
0
0
0
0
0
0
0
p
0
0.2
x1
(a)
0
0
0
0
0
0
0.4
x1
0.6
0.8
1
(b)
Fig. 7 The separatrix W s (pu ) (the bold solid curves) of the extinction and the recovery of the invasive
species by using ATYC model. The dash-dot (resp. dotted) curve shows the nullcline for x1 (t) = 0 (resp.
x2 (t) = 0). The directed solid (resp. dash-dot) curves are trial solutions, in which the dash-dot ones
converge to the recovery equilibrium ps , whereas the solid ones converge to the extinction equilibrium p0
shows that, at the LP, if x0 ≤ 0.1, there is a nearly linear relationship between α and
x0 ; if x0 > 0.1, this relationship becomes highly nonlinear.
Consider the case α > L P. Let W s (pu ) denote the stable manifold at pu . Figure 7
shows the phase plane of (5.2) with different values of α as x0 = 0.05. If the initial
position (x1 (0), x2 (0))T is below W s (pu ), the solution will approach p0 (which is
illustrated by the directed solid curves in Fig. 7). This demonstrates extinction of the
invasive species. Otherwise, if the initial (x1 (0), x2 (0))T is above W s (pu ), the solution
will converge to ps (which is illustrated by the directed dash-dot curves in Fig. 7). This
shows the recovery of the invasive fish species. Thus, W s (pu ) serves as the separatrix
that separates extinction from the recovery. Moreover, if α increases, Fig. 7 shows
that W s (pu ) is moving towards an L-shaped curve {(x1 , x2 ) : x1 = x0 , x2 = 0}. The
results indicate that: (1) the basin of attraction of the ES expands as α goes down to
the limit point L P; (2) the basin of attraction of the ES dramatically diminishes as
α increases. However, compared to the reduced TYC model (5.2), if α L P, the
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Analysis of the TYC eradication strategy for an invasive species
1749
wild-type invasive fish can only be established if the initial female population size is
above the critical value f 0 .
6 Turing instability
In this section, we study the possibility of a Turing (1952) instability for the TYC
model generalized by incorporating diffusive spatial spread.
6.1 A reaction-diffusion model on a line
First, we consider a continuous region in the shape of a line by assuming that the
density difference of each fish population at different position causes diffusion. Let
θ ∈ [0, 1] denote the position variable on a line. Assume that γi > 0 (1 ≤ i ≤ 4) is
the diffusion constant of f, m, s and r fish species. Then the generalized system of
(5.11) with inclusion of diffusive spatial spread takes the form:
∂ x1
∂τ
∂ x2
∂τ
∂ x3
∂τ
∂ x4
∂τ
= αx1 x2 L̂ − x1 + D̃1
∂ 2 x1
,
∂θ 2
= α(x1 x2 + x2 x4 + 2x1 x3 ) L̂ − x2 + D̃2
= α(x2 x4 + 2x3 x4 ) L̂ − x3 + D̃3
= ν − x4 + D̃4
∂ 2 x4
,
∂θ 2
∂ 2 x3
,
∂θ 2
∂ 2 x2
,
∂θ 2
(6.1)
where D̃i = γi /δ for 1 ≤ i ≤ 4 and ν = μ/(δ K ). We linearize (6.1) at the steady
state, p := (x1∗ , x3∗ , x3∗ , x4∗ )T , of the system (6.1) in the absence of diffusion. Set
xi = xi∗ + δxi , (i = 1, 2, 3, 4).
(6.2)
Let Z = (δx1 , δx2 , δx3 , δx4 )T . The associated linearized system in vector form is
given by
∂2
∂Z
= MZ + D̃ 2 Z,
∂τ
∂θ
(6.3)
where M is the Jacobian matrix of the associated ODE system evaluated at p, and
⎡
D̃1
⎢0
D̃ = ⎢
⎣0
0
0
D̃2
0
0
0
0
D̃3
0
⎤
0
0 ⎥
⎥.
0 ⎦
D̃4
123
1750
X. Wang et al.
Consider the eigenvalue problem
−υθθ (θ ) = σ υ(θ ), θ ∈ (0, 1),
θ = 0, 1.
υθ (θ ) = 0,
(6.4)
One can easily verify that the eigenvalues σk = (kπ )2 ≥ 0 and the corresponding
eigenfunctions υk (θ ) = cos(kπ θ ). We consider the ansatz
δxi (t, θ ) = eρτ υ(θ )ξi , (1 ≤ i ≤ 4),
(6.5)
where υ is the solution of the eigenvalue problem (6.4), and ρ and ξi are constant.
Substituting (6.5) into (6.3) yields
(ρI4 + σ D̃)ξ = Mξ.
(6.6)
We are interested in whether there exists ρ such that Re(ρ) > 0.
First, we consider the case μ = 0. Solve (6.6). We find that the associated eigenvalues are given by
ρ1 = η − 1 − σ ( D̃1 + D̃2 )/2 +
ρ2 = η − 1 − σ ( D̃1 + D̃2 )/2 −
η2 + (σ ( D̃1 − D̃2 )/2)2 ,
η2 + (σ ( D̃1 − D̃2 )/2)2 ,
ρ3 = −(σ D̃3 + 1),
ρ4 = −(σ D̃4 + 1),
(6.7)
where η is defined in (5.13).
Proposition 6.1 Suppose μ = 0. Inclusion of diffusive spatial spread into model
(5.11) will not produce a Turing instability.
Proof Since ρ2 , ρ3 and ρ4 are all negative and have the same sign as the corresponding
eigenvalue of M, the only eigenvalue that could have a sign change is ρ1 . Thus, it
suffices to study the sign of ρ1 .
If 0 < φ < 1, then (5.11) has three equilibria: p0 , pu and ps , in which p0 and ps
are stable.
Case 1 p = p0 , which represents the ES of (5.11). Then x ∗ = 0 and η = 0. Hence
(6.7) yields
ρ1 = −(σ D̃1 + 1) < 0.
Case 2 p = ps , which corresponds to the RS of(5.11).
√
√
In this case, x ∗ = (1 + 1 − φ)/4 and η = − 1 − φ + 1 − φ /φ + 1/2. Thus
0 < φ < 1 implies η < 1/2. By 2η − 1 < 0 and η − 1 < 0, we find that
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Analysis of the TYC eradication strategy for an invasive species
1751
2 ρ1 ρ2 = η − 1 − σ ( D̃1 + D̃2 )/2 − η2 + (σ ( D̃1 − D̃2 )/2)2 ,
= −(2η − 1) + 2σ D̃1 D̃2 − λ(η − 1)( D̃1 + D̃2 ) > 0.
Then ρ2 < 0, which implies that ρ1 < 0.
If φ ≥ 1, (5.11) has a unique stable equilibrium p0 . By the analysis in the first case
of 0 < φ < 1, all the eigenvalues associated with p0 will keep the same sign.
Therefore, inclusion of diffusive spatial spread will stabilize the system (5.11), and
will not be able to produce a Turing instability.
Remark In fact, if (i) φ = 1 or (ii) 0 < φ < 1 and −(2η − 1) + 2σ D̃1 D̃2 − σ (η −
1)( D̃1 + D̃2 ) > 0, then inclusion of diffusive spatial spread can even stablize the
unstable steady-state solution of model (5.11).
Now consider the case μ > 0. Define λ̂1 = −1, λ̂2 = αx4∗ (1 − x3∗ − x4∗ ) − 1, λ̂3 =
2αx4∗ (1 − 2x3∗ − x4∗ ) − 1, λ̂4 = −1, and Γˆ = Γ /δ.
Proposition 6.2 Suppose μ > 0. Assume that one of three hypothesis (H1)–(H3)
holds. Inclusion of diffusive spatial spread into model (5.11) does not produce a
Turing instability.
Proof If (H1)–(H3) hold, then by Theorem 3.1, the steady-state solution of (5.11)
would take the form (0, 0, x3∗ , x4∗ )T . Moreover, if Γˆ ≥ 0, (5.11) has a unique stable
equilibrium p := (0, 0, 0, x4∗ )T . If Γˆ < 0, (5.11) will have two equilibria p and
q := (0, 0, x3∗ , x4∗ )T , for which p is unstable and q is locally stable. In both cases, by
(6.6), we find that
ρi = λ̂i − σ D̃i , (i = 1, 2, 3, 4).
(6.8)
Note that {λ̂i : i = 1, 2, 3, 4} is the spectrum of M. For each 1 ≤ i ≤ 4, at the stable
fixed point, λ̂i < 0, and hence ρi < 0 by (6.8). So, no Turing instability would be
able to be established at the stable equilibrium for both cases by adding diffusion into
model (5.11).
6.2 A stepping-stone type model: reaction and migration in a circle
Fish populations migrate between regions. To study the effect of fish migration on the
closed area, a “stepping-stone” type of model, which is continuous in time and discrete
in state space, is developed as follows: (a) N colonies are assumed to be arranged on a
circle, labeled by integers n = 1, 2, · · · , N with f n (t), m n (t), sn (t) and rn (t) denoting
the number of f, m, s and r fish in colony n at time t, respectively. In particular n
and mod(n, N ) represent the same colony. (b) Each fish species is assumed to have a
constant migration rate. Let ζ1 , ζ2 , ζ3 and ζ4 denote the migration rates for fish species
f, m, s and r , respectively. (c) The individuals of each species are assumed to move
between adjacent colonies. Specifically, the overall migration into colony n is assumed
to come from the two nearest neighbors n − 1 and n colonies; whereas the overall
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X. Wang et al.
migration out of colony n is assumed to go to two nearest neighbors n − 1 and n + 1
colonies. Taking f n as an example, the rate at which f n increase due to the migration
from nearest neighbors colony n − 1 and colony n + 1 is then ζ1 ( f n−1 + f n+1 ); the
rate at which f n decreases due to the migration to colonies n − 1 and n + 1 is then
ζ1 f n + ζ1 f n , where the first term (resp. the second term) describes the migration from
colony n to colony n − 1 (resp. colony n + 1). Thus, the model with the inclusion of
fish migration can be written as:
d fn
1
= f n m n β L n − δ f n + ζ1 ( f n−1 − 2 f n + f n+1 ) ,
dt
2
1
dm n
1
=
f n m n + rn m n + f n sn β L n − δm n
dt
2
2
+ ζ2 (m n−1 − 2m n + m n+1 ) ,
1
dsn
=
rn m n + rn sn β L n − δsn + +ζ3 (sn−1 − 2sn + sn+1 ) ,
dt
2
drn
= μ − δrn + ζ4 (rn−1 − 2rn + rn+1 ) ,
dt
(6.9)
where
K.
L n = 1 − f n + m n + sn + r n
Let α = β K /(2δ), τ = δt, ν = μ/(δ K ) and σi = ζi /δ (for i = 1, 2, 3, 4). Write
x1n = f n /K , x2n = m n /K , x3n = sn /K and x4n = rn /K . Non-dimensionalization of
(6.9) yields
d x1n
dτ
d x2n
dτ
d x3n
dτ
d x4n
dτ
= αx1n x2n L n − x1n + σ1 x1n−1 − 2x1n + x1n+1 ,
= α x1n x2n + x4n x2n + 2x1n x3n L n − x2n + σ2 x2n−1 − 2x2n + x2n+1 ,
= α x4n x2n + 2x4n x3n L n − x3n + σ3 x3n−1 − 2x3n + x3n+1 ,
= ν − x4n + σ4 x4n−1 − 2x4n + x4n+1 ,
(6.10)
where
L n = 1 − x1n + x2n + x3n + x4n .
If the introduction of r fish is removed, μ = 0 and hence ν = 0. In this case, let
Fn (X) denote the right hand side of the system (6.10) without migration terms. Solve
Fn (p) = 0. We find the equilibrium p of each colony given by (x ∗ , x ∗ , 0, 0)T .
If the system is not far away from the equilibrium, we consider small perturbations
(δx1n , δx2n , δx3n , δx4n )T of the equilibrium
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Analysis of the TYC eradication strategy for an invasive species
1753
x nj = x ∗ + δx nj , ( j = 1, 2),
x nj
=
δx nj ,
( j = 3, 4).
(6.11)
(6.12)
Under this linear approximation, the Eq. (6.10) can be rewritten as
d
n+1
n
δx nj =
,
M jk δxkn + σ j δx n−1
−
2δx
+
δx
j
j
j
dτ
4
(6.13)
k=1
for 1 ≤ j ≤ 4. Here M = (M jk ) = D X n Fn Xn = p
with Xn = (x1n , x2n , x3n , x4n )T .
To solve (6.13) in this case, one can use discrete Fourier transformation
δx nj
N
1 r
n
,
=
u j exp 2πri
N
N
(6.14)
r =1
where
u rj =
N
n=1
√
r , and i = −1.
δx nj exp − 2π ni
N
Let Drj = 4σi sin2 (ωr ) with ωr = πr/N . Substitue (6.14) into Eq. (6.13). We find
that
du rj
dτ
=
4
M jk u rk − Drj u rj , 1 ≤ j ≤ 4, 1 ≤ r ≤ N .
(6.15)
k=1
Given 1 ≤ r ≤ N , the characteristic equation of (6.15) is given by
det(λI4 − (M − Dr )) = 0,
with
⎡
D1r
⎢
0
Dr = ⎢
⎣0
0
0
D2r
0
0
0
0
D3r
0
(6.16)
⎤
0
0 ⎥
⎥.
0 ⎦
D4r
Solve (6.16).
η2 + (D1r − D2r /2)2 ,
λr2 = η − 1 − (D1r + D2r )/2 − η2 + ( D1r − D2r /2)2 ,
λr1 = η − 1 − (D1r + D2r )/2 +
λr3 = −(D3r + 1),
λr4 = −(D4r + 1),
with η = αx ∗ (1 − 3x ∗ ).
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X. Wang et al.
Compared to the continuous case in Sect. 6.1, the arguments are similar and the
associated results on the existence of Turing instability are the same. So the details
are omitted for the discrete case.
7 Conclusion and discussion
The analysis presented here is intended to inform details of the behavior of certain
limiting mathematical systems modeling the TYC strategy for the biological control
of some very specific invasive species. In this work, we study questions that haven’t
been addressed in (Gutierrez and Teem 2006; Gutierrez 2005; Parshad and Gutierrez
2010). Specifically, we study the equilibrium and the stability of model (2.1).We also
address the natural question of how long must the Trojan females be supplied in
order to cause the species to die out. Thus, we investigate what will happen if this
injection is turned off, after some time. We theoretically study this situation through
bifurcation analysis. We find that the wide-type invasive population can either go
extinct or recover. Moreover, extinction and recovery are essentially modulated by a
parameter α = β K /(2δ), which is defined by the ratio of the the overall birth rate
in terms of the maximal capacity of the ecosystem to two times the per capita death
rate. If α is below the bifurcation value, only extinction can take place; if α is above
the bifurcation value, both extinction and recovery can happen, and the separatrix of
extinction and recovery is the stable manifold at the saddle. In particular, the basin
attraction of the extinction state dramatically shrinks as α increases. Moreover, we
identify a theoretical condition for the eradication strategy to work. Additionally, to
account for the difficulty of finding females at low levels of the population size, we also
incorporate an Allee effect into the TYC model (2.1). Unlike the original model (2.1),
the results for the ATYC model (4.1) show that the wild-type females need not have
nearly as low population size as that of (2.1) at the optimal termination time, and the
remaining sex-reversed supermales are sufficient to eradicate the wild-type females.
In addition, we study the possibility of a Turing instability by adding diffusive spatial
spread into model (2.1). We find that the inclusion of diffusive spatial spread does not
give rise to a Turing instability, which would have suggested that the TYC eradication
strategy might be only partially effective, leaving a patchy distribution of the invasive
species.
We would like to point out that there are a number of interesting questions at this
point, that would make for interesting future investigations. One could explore the
optimal control strategy, in this context. For instance, assume that the injection rate
of trojan fish, μ, is a function of time. We could ask: what’s the optimal control
T
for μ(t) such that the eradication strategy works and 0 μ(t)dt is minimized for a
given amount of time T ? It is also of interest to conduct a bifurcation analysis and
consider the Allee effect and inhomogeneous diffusive spatial spread for the partial
differential equation model. Here the investigations may have to resort heavily to
numerical methods. Another future direction is to consider stochastic models to study
the probability of extinction of an invasive species in a finite time, since the results
of the deterministic model in (Parshad and Gutierrez 2010) reveal extinction of an
invasive species is always possible as time goes to infinity.
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Analysis of the TYC eradication strategy for an invasive species
1755
All in all, we hope that our results will be of use in further developing the TYC
theory. The theory, although limited to species with XY sex determination system that
is capable of producing viable progeny during the sex reversal process, is a possible
means to combat invasive species of this type.
Acknowledgments This work was supported by the NSF-REU program DMS-0850470. This publication
is based in part on work supported by Award No. KUS-C1-016-04, made by King Abdullah University of
Science and Technology (KAUST).
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