Analysis of the Trojan Y-Chromosome eradication strategy ...mboggess/RESEARCH/trojan.pdfAbstract...

J. Math. Biol. (2014) 68:1731–1756DOI 10.1007/s00285-013-0687-1 Mathematical Biology

Analysis of the Trojan Y-Chromosome eradicationstrategy 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 biocontrolmethod, has been proposed to eliminate invasive alien species. In this work, we ana-lyze the dynamical system model of the TYC strategy, with the aim of studying theviability of the TYC eradication and control strategy of an invasive species. In par-ticular, because the constant introduction of sex-reversed trojan females for all timeis not possible in practice, there arises the question: What happens if this injection isstopped 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 therecovery state and the extinction state in both the full model and a certain reducedmodel. In particular, we find a theoretical condition for the eradication strategy towork. Additionally, the consideration of an Allee effect and the possibility of a Turinginstability are also studied in this work. Our results show that: (1) with the inclusionof an Allee effect, the number of the invasive females is not required to be very low

X. Wang (B) · J. R. Walton · M. BoggessDepartment of Mathematics, Texas A & M University, College Station, TX 77843, USAe-mail: [email protected]

J. R. Waltone-mail: [email protected]

M. Boggesse-mail: [email protected]

R. D. ParshadDivision of Mathematical and Computer Sciences and Engineering, King Abdullah Universityof Science and Technology, 23955 Thuwal, Kingdom of Saudi Arabiae-mail: [email protected]

K. StoreyDepartment of Mathematics, Carleton College, Northfield, MN 55057, USAe-mail: [email protected]

123

1732 X. Wang et al.

when the introduction of the sex-reversed trojan females is stopped, and the remain-ing Trojan Y-Chromosome population is sufficient to induce extinction of the invasivefemales; (2) incorporating diffusive spatial spread does not produce a Turing insta-bility, which would have suggested that the TYC eradication strategy might be onlypartially 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 economiclosses are attributed to various invasive species (Harmful 1993; Myers et al. 2000).Some well-known examples of these species include the burmese python in southernregions of the United States, the cane toad in Australia, and the sea lamprey and roundgoby in the Great Lakes region in the northern United States (Gutierrez 2005). Thecane toad was brought into Australia from Hawaii in 1935 in order to control the canebeetle. These have multiplied rapidly since then, and are currently Australia’s worstinvasive species (Philips and Shine 2004). The sea lamprey entered the great lakesin the 1800s, through shipping canals (Bryan et al. 2005), and these predators havecaused a severe decline in lake fish populations, adversely affecting local fisheries. Theburmese 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. Theseinvaders have had quite a negative impact on the local ecology, and climatic factorsmight support their spread to a third of the United States (Rodda et al. 2005). Apartfrom being considered one of the greatest threats to biodiversity, invasive species causeagricultural 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 noconclusive 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 ofa feasible biological control. It is well-known that the size of a population can beseriously affected as a result of shifting the sex ratio (Hamilton 1976). The sterile insecttechnique (SIT), works via the introduction of large quantities of sterile males, intoa target population, to compete with fertile males in mating (Knipling 1955). Matingwith sterile males does not produce any progeny, and the goal is to try and shift the sexratio so that there is an abundance of males, leading to less and less number of femalesover time. The method is commonly used in Florida, to control crop damage viaMediterranean fruit flies (Klassen and Curtis 2005). This in essence, is the underlyingprinciple for a number of recently proposed models (Howard et al. 2004; Gutierrezand Teem 2006; Bax and Thresher 2009), that use genetic modification to controlinvasive fish species. Among these models, the Daughterless Carp (Bax and Thresher

123

Analysis of the TYC eradication strategy for an invasive species 1733

2009) and the Trojan Y-Chromosome (TYC) (Gutierrez and Teem 2006), have beenparticularly well studied. The first strategy aims at causing extinction via the release ofindividuals with chromosomal insertions of aromatase inhibitors. The latter strategytries to cause extinction via the constant release of sex-reversed supermales, denotedas 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 supermaleprogeny, leading to less and less number of females over time, and ultimately drivingthe female population to zero. This will thus lead to extinction of the population. Notethat the TYC strategy only works for a few species with XY 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 strategywas proposed by Gutierrez and Teem (2006). Parshad and Gutierrez (2010) rigorouslyproved the existence of a global attractor for the model derived in (Gutierrez andTeem 2006), with the inclusion of diffusion. In particular, they show that the attractorpossesses the extinction state, if the introduction of sex-reversed supermales is largeenough. However, these works do not explore the full structure of the attractor, via arigorous stability and bifurcation analysis. Changing the sex ratio of a population seemsdifficult to achieve in practice, there have been a number of studies to this end (Nagleret al. 2001). It has also been reported that environmental pressure for masculinizationwould lead to extinction of fish populations with a XY sex-determination system(Hurley et al. 2004). Masculinization by exposure to androgens has been reported inthe wild (Katsiadakiand et al. 2002), and is common in the salmon industry to producean all-male stock (Hunter and Donaldson 1983).

In order to develop successful physical and biological controls for exotic fishspecies, the dynamics of fish populations and eradication strategies constructed onthe basis of these dynamics have to be studied and further understood. The goal of thiswork is to pursue a number of questions, that are not addressed in (Gutierrez and Teem2006; Parshad and Gutierrez 2010). These earlier works in essence, show that extinc-tion 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: whathappens if the introduction of the feminized supermales is stopped? Can the fish pop-ulation possibly recover, from low levels, instead of going to the extinction state? Anyeradication strategy, with potential to be implemented into a management practice,cannot possibly sustain the constant introduction of genetically modified organismsinto a target population, particularly over a large spatial domain. Thus, a rigorous sta-bility analysis of the model, in order to understand the specific structure of the attractorhas to be performed, in the case this release rate μ > 0, and μ = 0. In particular, theextinction or recovery, and their connection with the range of μ, has to be rigorouslyidentified. This is the first contribution of the current work. Next, the Allee effect (awell-known phenomenon from population dynamics, defined as a positive correlationbetween population density and individual fitness) (Odum and Allee 1954; Stephensand Sutherlan 1999) is incorporated into the current TYC model. Turing instability isalso explored in the context of the TYC model.

This article is organized as follows. Section 2 introduces the TYC eradicationstrategy and the TYC model developed in (Gutierrez and Teem 2006). Section 3presents the equilibrium and stability analysis of the TYC model. The Allee effect is

123

1734 X. Wang et al.

studied in Sect. 4. The basin of attraction of the extinction state is demonstrated inSect. 5 in terms of a reduced two-dimensional TYC model, the full TYC model whenμ = 0, and a reduced two-dimensional TYC model with the inclusion of an Alleeeffect. In Sect. 6, we show that incorporating diffusive spatial spread does not giverise 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 Y-Chromosome (TYC) eradication strategy for invasive fish species. The idea behindthis strategy is to eliminate the wild-type female fish from a population, by introduc-ing 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-typefemales over time, ultimately leading to their eradication, and thus causing the pop-ulation to go extinct. The strategy relies on two facts: (1) the wild-type invasive fishspecies are required to have an XY sex-determination system, that is, sex is determinedby the presence of a Y chromosome in a fish species; (2) the sex change of geneticmales to females and of females to males can be made through chemical inducedgenetic manipulation, such as hormone treatments.

Gutierrez and Teem (2006) considered the Nile tilapia as an example. Their TYCmodel describes the dynamics of four phenotype/genotype variants of the invasivefish species: genotype XX females, genotype XY males, phenotype YY supermalesand phenotype sex-reversed YY females, denoted as f, m, s and r , respectively. Theeradication strategy works as follows. A sex-reversed trojan female (r), carrying twoY Chromosomes, is introduced into a wild-type invasive fish population comprisedof 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. Thehigh incidence of males decreases the sex ratio of females to males. Ultimately, thewild-type females would be eliminated from the population. As a consequence, thewild-type invasive population goes extinct.

It follows from the pedigree tree of the TYC model illustrated in Fig. 1, that theequation 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)

123


d f

dt= 1

2f mβL − δ f,

dm

dt=

(12

f m + 12

rm + f s)βL − δm,

ds

dt=

(12

rm + rs)βL − δs,

dr

dt= μ − δr, (2.1)

where f, m, s, r define the number of individuals in each associated class; positiveconstants β and δ represent the per capita birth and death rates, respectively; nonnega-tive 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 + rK

, (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 thesolution 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 rateμ > 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 − βK δμ + K δ3. (3.1)

If Γ ≥ 0, Eq. (2.1) has only one equilibrium that satisfies f ∗ = 0, namely,(0, 0, 0,

μ

δ

)T. Otherwise, if Γ < 0, Eq. (2.1) has two equilibria with f ∗ = 0.

One is(

0, 0, 0,μ

δ

)T, and the other one is

(0, 0, K

(1 − δ

2

μβ

)− μ

δ,μ

δ

)T. Clearly, if

f ∗ > 0, m∗ > 0 and r∗ > 0, then s∗ > 0, and s∗ = 0 is not a solution of biologicalmeaning.

123

1736 X. Wang et al.

Case 2 Assume f ∗ > 0.By direct computations, we find that the equilibrium satisfies

f ∗ = (r∗)2(r∗ + s∗)s∗(s∗ − r∗) ,

m∗ = 2r∗s∗

s∗ − r∗ ,

r∗ = μδ

, (3.2)

where s∗ satisfies

s(as3 + bs2 + cs + d) = 0, (3.3)with a = a(r∗) := r∗β, b = b(r∗) := 2β(r∗)2 − βKr∗ + δK , c = c(r∗) :=βK (r∗)2 − 2δKr∗, 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 andd > 0. Clearly, s∗ > r∗ > 0 is required for the equilibrium of model (2.1) to be in theregion Ω .

Let r±b =K

4±

√( K4

)2 − δK2β

and rc = 2δβ

. It is easy to verify that r±b and rc arethe nonzero roots of b(r∗) = 0 and c(r∗) = 0, respectively.

We define α = βK/2δ.Lemma 3.1 (a) If α = 4, r−b = rc = r+b , (b) If α > 4, then r−b < rc < r+b .Proof One can easily verify the result by direct calculation. ��For simplicity, we write Δ = 18abcd − 4b3d + (bc)2 − 4ac3 − 27(ad)2. In fact, Δis the determinant of a cubic equation as3 + bs2 + cs + d = 0. The three cases forthis determinant are well known.

Proposition 3.1 Assume that Δ > 0.

(a) If α > 4, then Eq. (3.3) has two positive roots forμ

δ< r+b , and no positive roots

forμ

δ≥ r+b .

(b) If α ≤ 4, then Eq. (3.3) has two positive roots for μδ

< rc, and no positive roots

forμ

δ≥ rc.

Proof We define the term in parentheses of Eq. (3.3) to be

g(s) = as3 + bs2 + cs + d. (3.4)By a > 0 and Δ > 0, Eq. (3.4) has three distinct real roots. Moreover, none of theroots can be zero because d �= 0.Case (a) α > 4. By Lemma 3.1, we have r−b < rc < r

+b .

(1) Assume r∗ = μ/δ ∈ (0, r−b ). Then b > 0 and c < 0. The signs of the coefficientsof g(s) and g(−s) are

123


sign(a, b, c, d) = (+,+,−,+), (3.5)

and

sign(−a, b,−c, d) = (−,+,+,+), (3.6)

respectively. So there are 2 sign changes in g(s) and 1 sign change in g(−s). ByDescartes’ rule of signs, we see that g(s) = 0 has either 2 or 0 positive roots andexactly 1 negative root. Since g(s) = 0 has three nonzero real roots, the rootshave to be comprised of 2 positive ones and 1 negative one.

(2) Assume r∗ ∈ [r−b , r+b ). If r∗ = r−b , then b = 0 and c < 0. If r∗ ∈ (r−b , rc), thenb < 0 and c < 0. If r∗ = rc, then b < 0 and c = 0. If r∗ ∈ (rc, r+b ). then b < 0and c > 0. By similar arguments, we can show that g(s) has two positive zeros.

(3) Assume r∗ ∈ [r+b ,∞). If r∗ = r+b , then b = 0 and c > 0. If r∗ > r+b , then b > 0and c > 0. Since there are no changes of sign in the coefficients, there are nopositive 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.(a) If α > 4, then Eq. (3.3) has either zero or two positive roots for

μ

δ< r+b , and no

positive roots forμ

δ≥ r+b .

(b) If α ≤ 4, then Eq. (3.3) has either zero or two positive roots for μδ

< rc, and no

positive roots forμ

δ≥ rc.

Proof If Δ = 0, Eq. (3.4) has a multiple root and all its roots are real. The resultsfollow 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. Theresults 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

Jq =

⎡⎢⎢⎢⎣

β

2m∗L∗ − ξ − δ β

2f ∗L∗ − ξ −ξ −ξ

(m∗/2 + s∗)βL∗ − η ( f ∗/2 + r∗/2)βL∗ − η − δ β f ∗L∗ − η m∗βL∗/2 − η−κ βr∗L∗/2 − κ βr∗L∗ − κ − δ (m∗/2 + s∗)βL∗ − κ0 0 0 −δ

⎤⎥⎥⎥⎦

(3.7)

123

1738 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 bewritten as:

(λ + δ)[λ −

(β

2r∗L∗ − δ

)] [λ −

(βr∗

(L∗ − s

∗

2K

)−δ

)](λ + δ)=0, (3.8)

and the corresponding eigenvalues are given by

λ1 = −δλ2 = β

2r∗L∗ − δ

λ3 = βr∗(L∗ − s

∗

K

) − δλ4 = −δ. (3.9)

Hence, the eigenvalues associated with p := (0, 0, 0, μ/δ)T are λ1 = −δ, λ2 =βr∗L∗/2 − δ = −Γ/(2δ2 K ) − δ/2, λ3 = βr∗L∗ − δ = −Γ/(δ2 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 −δ2

μβ

)− μ

δ,μ

δ

)T. One can easily verify that the eigenvalues associated with q are

λ1 = −δ, λ2 = −δ/2, λ3 = −βr∗s∗/K and λ4 = −δ. These eigenvalues are allnegative, 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.

3.1.3 A condition for the eradication strategy

Let

(H1) : � ≥ 0, α > 4, and μ/δ ≥ r+b ; (3.10)(H2) : � ≥ 0, α ≤ 4, and μ/δ ≥ rc; (3.11)(H3) : � < 0. (3.12)

By Propositions 3.1 and 3.3, if one of three hypothesis (H1)–(H3) is valid, the TYCmodel 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. Alto-gether 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 erad-ication strategy works. Moreover, if Γ < 0, the TYC model (2.1) has two equilibria:

123


p := (0, 0, 0, μ/δ)T and q :=(

0, 0, K(

1− δ2

μβ

)− μ

δ,μ

δ

)T, and p is locally unstable

whereas q is locally stable; if Γ ≥ 0, then the system (2.1) has only one equilibrium pand it is locally stable. In either case, the stable equilibrium represents the eliminationstate 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 caneliminate the wild-type females from the population. However, it is not practical tokeep the introduction constant for all time. So, a natural question is: what happens ifthe 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/βK . Clearly, φ > 0 since β, K and δ are all assumedto be positive. Let f ∗± =

K

4(1 ± √1 − φ). If φ < 1, the TYC model (2.1) has three

distinct equilibria for which f ∗ = f ∗+, f ∗− and 0, respectively. If φ = 1, the modelhas two equilibria and f ∗ = K

4and 0, respectively. Otherwise if φ > 1, the system

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

(λ + δ)3

(λ − (2γ − δ)

)= 0, (3.14)

where γ = 0, or γ = γ± := β2

f ∗±(

1 − 3 f∗±

K

). The associated eigenvalues are λi =

−δ < 0, for i = 1, 2, 3, and λ4 = −δ, 2γ− − δ or 2γ+ − δ. Therefore, the sign of λ4determines the local stability of the equilibrium. In particular, at (0, 0, 0, 0)T , γ = 0and hence λi = −δ < 0 for all 1 ≤ i ≤ 4.

In addition, it follows from direct calculation that

2γ± − δ = −2δφ

(1 − φ ± √1 − φ

). (3.15)

Then φ ≤ 1 yields 2γ+ − δ < 0 and 2γ− − δ > 0. It demonstrates that the equi-librium (0, 0, 0, 0)T (which represents the elimination state) and ( f ∗+, f ∗+, 0, 0)T arelocally stable (which represents the recovery state), whereas ( f ∗−, f ∗−, 0, 0)T is locallyunstable, and is a saddle.

123

1740 X. Wang et al.

4 The Allee effect

The (component) Allee effect is a positive relationship between any measurable com-ponent of individual fitness and population size (Odum and Allee 1954; Stephens andSutherlan 1999). Population-dynamic consequences of the Allee effect are of primaryimportance in conservation biology and other fields of ecology.

Due to the difficulty of finding mates at low population sizes, we introduce theAllee effect into the mathematical model of the TYC eradication strategy. A reasonablemodel (2.1) with the inclusion of an Allee effect takes the form:

d f

dt= 1

2f m

(f

f0− 1

)βL − δ f,

dm

dt=

(1

2f m

(f

f0− 1

)+ 1

2rm

(r

r0− 1

)+ f s

(f

f0− 1

))βL − δm,

ds

dt=

(12

rm + rs)( r

r0− 1

)βL − δs,

dr

dt= μ − δr,

L = 1 − f + m + s + rK

, (4.1)

where f0 (resp. r0) is a critical value of f fish (resp. r fish), which is used to characterizethe minimal group size to find mates, rear offspring, search for food and/or fend offattacks from predators. Mathematically, we assume that f0 � K , and r0 = f0.We denote this model (4.1) by ATYC model. Here our use of the cubic form forincorporating an Allee effect is for illustrative purposes only in the same spirit as itsuse in (Lewis and Kareiva 1993), and in the classical KPP equation (Kolmogorov etal. 1937).

The eradication strategy works in the way that the wild-type female population canbe 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 trojanfish is removed. Now the question is when would be the optimal termination time forsuch a removal? Here, by the “optimal termination time”, we mean the minimal timethat is required to switch off the injection of the trojan fish so that the wild-type fishspecies 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 thenumber of the wild-type females, f , is extremely low to guarantee that the wild-typefish will not recover. In this case, τ0 ≈ 58 (time unit), and f (τ0) < 1, which indeedprovides no information for biological implementation. The middle (resp. the bottom)panel in Fig. 2 displays the associated results for the ATYC models with f0 = 0.01K(resp. f0 = 0.02K ). Specifically, if f0 = 0.01K , τ0 ≈ 34 and f (τ0) ≈ 21 (whichis about 0.07K ); if f0 = 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 optimaltermination time τ0 needs not be very low for the ATYC model; (2) the remaining

123


0 50 100 150 200 250 3000

50

100

150

200

250

300

time

XX female (f)XY male (m)YY super male (s)YY female (r)

0 50 100 150 200 250 3000

50

100

150

200

250

300

time


0 50 100 150 200 250 3000

50

100

150

200

250

300

time


0 50 100 150 200 250 3000

50

100

150

200

250

300

time


0 50 100 150 200 250 3000

50

100

150

200

250

300

time


0 50 100 150 200 250 3000

50

100

150

200

250

300

time


(a) (b)

(c) (d)

(e) (f)

Fig. 2 The Allee effect is illustrated by comparing the solutions of the TYC model and those of ATYCmodel 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 f0 = 3 and f0 = 6) at different termination time of t0. Here μ = 20 ift ≤ 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 afterτ0.

5 The basin of attraction of the extinction state of a wild-type species whenµ = 0

5.1 The reduced two-dimensional TYC model

The equilibrium analysis in Sect. 3.2.1 shows that both r and s would exponentiallydecay to 0 if μ = 0. If r = s = 0 in the original model, (2.1) gives a reducedtwo-dimensional system as follows:

d f

dt= 1

2f mβ

(1 − f + m

K

)− δ f,

dm

dt= 1

2f mβ

(1 − f + m

K

)− δm. (5.1)

Nondimensionalize model (5.1). Let x1 = f/K , x2 = m/K and τ = δt . Recall thatα = βK/(2δ). We get

dx1dτ

= αx1x2(1 − x1 − x2) − x1,dx2dτ

= αx1x2(1 − x1 − x2) − x2. (5.2)

It can be seen from the bifurcation diagram for model (5.2) which is displayedin Fig. 3, if α < 8 (i.e., φ = 16δ/(βK ) = 8/α > 1), there is only one stableequilibrium. If α = 8 (i.e., φ = 1), there are two equilibria: one is stable and the otherone is unstable. If α > 8 (i.e., φ < 1), there are three equilibria: one is unstable and theother two are stable. Focus on the most interesting case where α > 8. One can verifythat the three equilibria are: p0 = (0, 0), pu = ((1 −

√1 − φ)/4, (1 − √1 − φ)/4),

and ps = ((1 +√

1 − φ)/4, (1 + √1 − φ)/4). In particular, p0 represents the ES,and ps represents the recovery state (RS). Let (x

∗1 , x

∗2 ) denote the equilibrium of (5.2).

Fig. 3 Bifurcation diagram forthe reduced two-dimensionalmodel (5.2). The stableequilibrium and the unstableequilibrium are displayed by thesolid curve and the dash-dotcurve, respectively

0 5 10 15 20 25

0

0.1

0.2

0.3

0.4

α

x 1

123


Write η = αx∗1 (1−3x∗1 ). The eigenvalues of the system (5.2) are given by λ1 = 2η−1and 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 thesaddle pu . Let

Rθ =[

cos(θ) − sin(θ)sin(θ) cos(θ)

](5.3)

denote the rotation matrix. Consider the coordinate transformation

(Δx1Δx2

)= Rπ/4

(US

), (5.4)

for which Δx1 = x1 − x∗1 and Δx2 = x2 − x∗2 . This yields

d

dt

(US

)=

(λ1 00 λ2

) (US

)+

(F(U, S)0

), (5.5)

where

F(U, S) = −√2αx∗(U 2 + S2) + α(1 − 4x∗)(U 2 − S2)/√2 − α(U 2 − S2)U.(5.6)

We now look for the curve on the (U, S) phase plane such that it forms the separatrixbetween the basis of attraction for the ES and that for the RS. The separatrix indeedis 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

λ2S(h(S) + Sh′(S)) = λ1Sh(S) + F(Sh(S), S). (5.7)

In view of Eq. (5.7) and h(0) = 0, we have

h(S) = 1λ2

S−nS∫

0

ξn−2 F(ξh(ξ), ξ)dξ, (5.8)

where n = 1 − λ1/λ2. It is clear that n > 1 because λ1 > 0 > λ2. Substituteλ1 = 2η − 1 and λ2 = −1 into the expression of n. We get n = 2η. It then followsfrom the definition of F(U, S) in (5.6) that

F(U, S) = c20U 2 + c02S2 + c30U 3 + c12U S2, (5.9)

for which c20 = α(1 − 6x∗1 )/√

2, c02 = α(−1 + 2x∗1 )/√

2, c30 = −α and c12 = α.Substitute (5.9) and λ2 = −1 into (5.8). We have

123

1744 X. Wang et al.

x1

x 2

p0

pu

ps

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1x1’(t) = 0

x2’(t) = 0

Ws(pu)

(a)

x1

x 2

p0

pu

ps

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1x1’(t) = 0

x2’(t) = 0

Ws(pu)

(b)

Fig. 4 The separatrix W s (pu) of the extinction and the recovery of the invasive fish species f , whichare displayed in bold solid curves. The dashed (resp. dotted) curve is the nullcline for x ′1(t) = 0 (resp.x ′2(t) = 0). The solid and dashed curves are trial solutions of (5.2). The dashed ones converge to therecovery equilibrium ps , whereas the solid ones converge to the extinction equilibrium p0

h(S) = −S1∫

0

θn(c20(h(Sθ))

2 + c02 + c30Sθ(h(Sθ))3 + c12Sθh(Sθ))dθ.

(5.10)

In general, (5.10) has no closed-form analytical solution. We have to rely on thenumerical method to evaluate it (Atkinson 1992, 1997). Let W s(pu) denote the stablemanifold at pu . Figure 4 shows W

s(pu) with different values of the parameter α. Ifthe initial position (x1(0), x2(0))T is below W s(pu), the solution will converges top0 (which is illustrated by the directed solid curves in Fig. 4). This demonstrates theextinction of the invasive species. Otherwise, if the initial (x1(0), x2(0))T is aboveW s(pu), the solution will converge to ps (which is illustrated by the directed dashedcurves 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 αincreases, Fig. 4 shows that the separatrix W s(pu) is moving towards an L-shapedcurve which comprises two segments of axes in the first quadrant restricted to the unitsquare [0, 1] × [0, 1]. Thus, the results show that: (1) the basin of attraction of the ESexpands as α goes down to the bifurcation point 8; (2) the basin of attraction of the ESdramatically decreases as α goes up and becomes unbounded. Moreover, if α � 8, thewild-type invasive fish can always be established, as along as their initial populationis 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 invasivefish species will die off or recover after the removal of the introduction of sex-reversedsupermales, 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


dx1dτ

= αx1x2 L̂ − x1,dx2dτ

= α(x1x2 + x2x4 + 2x1x3)L̂ − x2,dx3dτ

= α(x2x4 + 2x3x4)L̂ − x3,dx4dτ

= −x4, (5.11)

where L̂ = 1 − ∑4i=1 xi . Define X = (x1, x2, x3, x4)4 and F to be vector of the righthand sides of Eq. (5.11). Then Eq. (5.11) can be written as:

dXdτ

= F(X). (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

M = DF(p∗) =

⎡⎢⎢⎣

η − 1 η −B −Bη η − 1 2A − B η0 0 −1 A0 0 0 −1

⎤⎥⎥⎦ , (5.13)

where η = A − B with A = αx∗ L̂∗, B = α (x∗)2 and L̂∗ = 1 − 2x∗. If x∗ = 0, thenM = −I4 where I4 represents the 4 × 4 identity matrix. If x∗ �= 0, then η = A − B =αx∗(1 − 3x∗) > 0. Let

T =

⎡⎢⎢⎣

T11 T12 T13 −T23−T11 T22 T23 −T230 T32 T33 00 0 1 0

⎤⎥⎥⎦ ,

where T11 = −A2, T12 = (2AB − 3A2)/(2η), T13 = (−7A2 + 8AB − 2B2 +2ηB)/(4η2), T22 = A2/(2η), T23 = A2/(4η2), T33 = (2A − B − η)/(2η), andT32 = A. Then by the coordinate transformation X = p + TY, Eq. (5.12) yields

dYdt

= JY + G(Y), (5.14)

where J is the Jordan canonical form of M with

J =

⎡⎢⎢⎣

−1 1 0 00 −1 1 00 0 −1 00 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 Jordanform (5.13) indicates that pu is hyperbolic of saddle type. Thus, there exists a solutionflow of equation (5.14), �(t; ξ1, ξ2, ξ3) ∈ C1([0,∞), U ), for some neighborhoodU ⊂ R3, and limt→∞ �(t; ξ1, ξ2, ξ3) = 0. Specifically, for any C = (c1, c2, c3, 0)Twith (c1, c2, c3)T ∈ U, �(t, ·) is given by

�(τ ; C) = �s(τ )C +τ∫

0

�s(τ − s)G(�(s))ds −∞∫

τ

�u(τ − s)G(�(s))ds,

where

�s(τ ) =

⎡⎢⎢⎣

e−τ τe−τ 12τ2e−τ 0

0 e−τ τe−τ 00 0 e−τ 00 0 0 0

⎤⎥⎥⎦ ,

and

�u(τ ) =

⎡⎢⎢⎣

0 0 0 00 0 0 0.eps0 0 0 00 0 0 eσ4τ

⎤⎥⎥⎦ .

Then system (5.14) processes a 3-dimensional local stable manifold at the origin 0,

W sloc(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 = (n1, n2, n3, n4)T with n1 = n2 = 1, n3 = −(T12 + T22)/T32 and

n4 =(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 thesex-reversed supermales r is switched off. Since pu is hyperbolic of saddle type, thereexists a neighborhood U0 such that W sloc(0) is well-defined and the following resultshold: (1) if X(t0) satisfies T−1(X(t0) − pu) ∈ U0 and T−1(X(t0) − pu) · n > 0, thesolution of Eq. (5.11) will converge to ps , which implies that the invasive fish speciesf and m will recover after the removal of r (see the red curves in Fig. 5); (2) whereasif 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 oftrojan females is stopped depends on the value of the parameter α and the initial stateof the trajectories. Specifically, if α is a constant, the stable manifold of pu providesa separatrix of extinction and recovery. If the initial state is identical, increasing αwill result in the basin of the attraction of the extinction shrinking whereas that of therecovery expanding (see Fig. 5).

123


Fig. 5 Projection of some trajectories of the TYC model (5.11) onto the x1x2 plane. The red (resp. blue)curves represents recovery (resp. extinction) of the wild-type invasive species. The associated initials ofthese trajectories are in a close neighborhood of the saddle fixed point pu . The black curves show thesolutions of (5.11) that are very close to the projection of the separatrix of recovery and extinction onto thex1x2 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 reducedtwo-dimensional system

d f

dt= 1

2f mβL

(f

f0− 1

)− δ f,

dm

dt= 1

2f mβL

(f

f0− 1

)− δm. (5.16)

Let x1 = f/K , x2 = m/K , x0 = f0/K and τ = δt . Nondimensionalizing model(5.16) gives

dx1dτ

= αx1x2(1 − x1 − x2)(x1/x0 − 1) − x1,dx2dτ

= αx1x2(1 − x1 − x2)(x1/x0 − 1) − x2. (5.17)

Bifurcation diagrams for the reduced model (5.17) are displayed in Fig. 6. Figure6a 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) changesat the limit point, L P . There is a unique stable equilibrium point p0 = (0, 0)T ifα < L P (α ≈ 1.6 as x0 = 0.05). At α = L P , there are two equilibria and only thelower 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 asaddle, and both p0 (representing the ES) and ps (repenting the RS) are stable. Thus, asaddle-node bifurcation occurs at L P . If x0 varies, Fig. 6 b displays a two-parameterbifurcation, which illustrates how α changes as x0 varies at the LP. In particular, it

123

1748 X. Wang et al.

0 5 10 15 20 25

0

0.1

0.2

0.3

0.4

0.5

α

x 1

LP

0 20 40 60 80 1000

0.1

0.2

0.3

0.4

α

x 0

(a) (b)

Fig. 6 Bifurcation diagram for the reduced two-dimensional model (5.17). a x1 as a function of α withx0 = 0.05. The stable equilibrium (resp. unstable equilibrium) is displayed by the solid curve (resp. thedash-dot curve). b Two-parameter bifurcation showing α as a unction of x0 at the limit point (LP)

00

0

0

0

0

0 0

0

0

0

x1

x 2

p0

pu

ps

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1x1’(t) = 0

x2’(t) = 0

ws(pu)

(a)

0

00

0

0

0

0

00

0

0

0 000

0

0

0

0

0

0

x1

x 2

p0

pu

ps

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1x1’(t) = 0

x2’(t) = 0

ws(pu)

(b)

Fig. 7 The separatrix W s (pu) (the bold solid curves) of the extinction and the recovery of the invasivespecies by using ATYC model. The dash-dot (resp. dotted) curve shows the nullcline for x ′1(t) = 0 (resp.x ′2(t) = 0). The directed solid (resp. dash-dot) curves are trial solutions, in which the dash-dot onesconverge 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 andx0; 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 7shows the phase plane of (5.2) with different values of α as x0 = 0.05. If the initialposition (x1(0), x2(0))T is below W s(pu), the solution will approach p0 (which isillustrated by the directed solid curves in Fig. 7). This demonstrates extinction of theinvasive species. Otherwise, if the initial (x1(0), x2(0))T is above W s(pu), the solutionwill converge to ps (which is illustrated by the directed dash-dot curves in Fig. 7). Thisshows the recovery of the invasive fish species. Thus, W s(pu) serves as the separatrixthat separates extinction from the recovery. Moreover, if α increases, Fig. 7 showsthat W s(pu) is moving towards an L-shaped curve {(x1, x2) : x1 = x0, x2 = 0}. Theresults indicate that: (1) the basin of attraction of the ES expands as α goes down tothe limit point L P; (2) the basin of attraction of the ES dramatically diminishes asα increases. However, compared to the reduced TYC model (5.2), if α � L P , the

123


wild-type invasive fish can only be established if the initial female population size isabove the critical value f0.

6 Turing instability

In this section, we study the possibility of a Turing (1952) instability for the TYCmodel 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 thedensity difference of each fish population at different position causes diffusion. Letθ ∈ [0, 1] denote the position variable on a line. Assume that γi > 0 (1 ≤ i ≤ 4) isthe 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

= αx1x2 L̂ − x1 + D̃1 ∂2x1

∂θ2,

∂x2∂τ

= α(x1x2 + x2x4 + 2x1x3)L̂ − x2 + D̃2 ∂2x2

∂θ2,

∂x3∂τ

= α(x2x4 + 2x3x4)L̂ − x3 + D̃3 ∂2x3

∂θ2,

∂x4∂τ

= ν − x4 + D̃4 ∂2x4

∂θ2, (6.1)

where D̃i = γi/δ for 1 ≤ i ≤ 4 and ν = μ/(δK ). We linearize (6.1) at the steadystate, p := (x∗1 , x∗3 , x∗3 , x∗4 )T , of the system (6.1) in the absence of diffusion. Set

xi = x∗i + δxi , (i = 1, 2, 3, 4). (6.2)

Let Z = (δx1, δx2, δx3, δx4)T . The associated linearized system in vector form isgiven by

∂Z∂τ

= MZ + D̃ ∂2

∂θ2Z, (6.3)

where M is the Jacobian matrix of the associated ODE system evaluated at p, and

D̃ =

⎡⎢⎢⎣

D̃1 0 0 00 D̃2 0 00 0 D̃3 00 0 0 D̃4

⎤⎥⎥⎦ .

123

1750 X. Wang et al.

Consider the eigenvalue problem

{−υθθ (θ) = συ(θ), θ ∈ (0, 1),υθ (θ) = 0, θ = 0, 1. (6.4)

One can easily verify that the eigenvalues σk = (kπ)2 ≥ 0 and the correspondingeigenfunctions υk(θ) = cos(kπθ). 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 eigen-

values are given by

ρ1 = η − 1 − σ(D̃1 + D̃2)/2 +√

η2 + (σ (D̃1 − D̃2)/2)2,ρ2 = η − 1 − σ(D̃1 + D̃2)/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 correspondingeigenvalue of M , the only eigenvalue that could have a sign change is ρ1. Thus, itsuffices to study the sign of ρ1.

If 0 < φ < 1, then (5.11) has three equilibria: p0, pu and ps , in which p0 and psare 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

123


ρ1ρ2 =(η − 1 − σ(D̃1 + D̃2)/2

)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 theunstable steady-state solution of model (5.11).

Now consider the case μ > 0. Define λ̂1 = −1, λ̂2 = αx∗4 (1− x∗3 − x∗4 )−1, λ̂3 =2αx∗4 (1 − 2x∗3 − x∗4 ) − 1, λ̂4 = −1, and Γ̂ = 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 aTuring instability.

Proof If (H1)–(H3) hold, then by Theorem 3.1, the steady-state solution of (5.11)would take the form (0, 0, x∗3 , x∗4 )T . Moreover, if Γ̂ ≥ 0, (5.11) has a unique stableequilibrium p := (0, 0, 0, x∗4 )T . If Γ̂ < 0, (5.11) will have two equilibria p andq := (0, 0, x∗3 , x∗4 )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 stablefixed point, λ̂i < 0, and hence ρi < 0 by (6.8). So, no Turing instability would beable to be established at the stable equilibrium for both cases by adding diffusion intomodel (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 theclosed area, a “stepping-stone” type of model, which is continuous in time and discretein state space, is developed as follows: (a) N colonies are assumed to be arranged on acircle, labeled by integers n = 1, 2, · · · , N with fn(t), mn(t), sn(t) and rn(t) denotingthe number of f, m, s and r fish in colony n at time t , respectively. In particular nand mod(n, N ) represent the same colony. (b) Each fish species is assumed to have aconstant migration rate. Let ζ1, ζ2, ζ3 and ζ4 denote the migration rates for fish speciesf, m, s and r , respectively. (c) The individuals of each species are assumed to movebetween adjacent colonies. Specifically, the overall migration into colony n is assumedto come from the two nearest neighbors n − 1 and n colonies; whereas the overall

123

1752 X. Wang et al.

migration out of colony n is assumed to go to two nearest neighbors n − 1 and n + 1colonies. Taking fn as an example, the rate at which fn increase due to the migrationfrom nearest neighbors colony n − 1 and colony n + 1 is then ζ1( fn−1 + fn+1); therate at which fn decreases due to the migration to colonies n − 1 and n + 1 is thenζ1 fn + ζ1 fn , where the first term (resp. the second term) describes the migration fromcolony n to colony n − 1 (resp. colony n + 1). Thus, the model with the inclusion offish migration can be written as:

d fndt

= 12

fnmnβLn − δ fn + ζ1 ( fn−1 − 2 fn + fn+1) ,dmndt

=(

1

2fnmn + 1

2rnmn + fnsn

)βLn − δmn

+ ζ2 (mn−1 − 2mn + mn+1) ,dsndt

=(

1

2rnmn + rnsn

)βLn − δsn + +ζ3 (sn−1 − 2sn + sn+1) ,

drndt

= μ − δrn + ζ4 (rn−1 − 2rn + rn+1) , (6.9)

where

Ln = 1 −(

fn + mn + sn + rn)/

K .

Let α = βK/(2δ), τ = δt, ν = μ/(δK ) and σi = ζi/δ (for i = 1, 2, 3, 4). Writexn1 = fn/K , xn2 = mn/K , xn3 = sn/K and xn4 = rn/K . Non-dimensionalization of(6.9) yields

dxn1dτ

= αxn1 xn2 Ln − xn1 + σ1(

xn−11 − 2xn1 + xn+11),

dxn2dτ

= α(

xn1 xn2 + xn4 xn2 + 2xn1 xn3

)Ln − xn2 + σ2

(xn−12 − 2xn2 + xn+12

),

dxn3dτ

= α(

xn4 xn2 + 2xn4 xn3

)Ln − xn3 + σ3

(xn−13 − 2xn3 + xn+13

),

dxn4dτ

= ν − xn4 + σ4(

xn−14 − 2xn4 + xn+14), (6.10)

where

Ln = 1 −(

xn1 + xn2 + xn3 + xn4).

If the introduction of r fish is removed, μ = 0 and hence ν = 0. In this case, letFn(X) denote the right hand side of the system (6.10) without migration terms. SolveFn(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(δxn1 , δx

n2 , δx

n3 , δx

n4 )

T of the equilibrium

123


xnj = x∗ + δxnj , ( j = 1, 2), (6.11)xnj = δxnj , ( j = 3, 4). (6.12)

Under this linear approximation, the Eq. (6.10) can be rewritten as

d

dτδxnj =

4∑k=1

M jkδxnk + σ j

(δxn−1j − 2δxnj + δxn+1j

), (6.13)

for 1 ≤ j ≤ 4. Here M = (M jk) = DXn Fn∣∣∣Xn=p

with Xn = (xn1 , xn2 , xn3 , xn4 )T .To solve (6.13) in this case, one can use discrete Fourier transformation

δxnj =1

N

N∑r=1

urj exp(

2πrin

N

), (6.14)

where

urj =N∑

n=1δxnj exp

(− 2πni r

N

), and i = √−1.

Let Drj = 4σi sin2(ωr ) with ωr = πr/N . Substitue (6.14) into Eq. (6.13). We findthat

durjdτ

=4∑

k=1M jku

rk − Drj urj , 1 ≤ j ≤ 4, 1 ≤ r ≤ N . (6.15)

Given 1 ≤ r ≤ N , the characteristic equation of (6.15) is given bydet(λI4 − (M − Dr )) = 0, (6.16)

with

Dr =

⎡⎢⎢⎣

Dr1 0 0 00 Dr2 0 00 0 Dr3 00 0 0 Dr4

⎤⎥⎥⎦ .

Solve (6.16).

λr1 = η − 1 − (Dr1 + Dr2)/2 +√

η2 + ((Dr1 − Dr2)/2)2,

λr2 = η − 1 − (Dr1 + Dr2)/2 −√

η2 + ((Dr1 − Dr2)/2)2,

λr3 = −(Dr3 + 1),λr4 = −(Dr4 + 1),

with η = αx∗(1 − 3x∗).

123

1754 X. Wang et al.

Compared to the continuous case in Sect. 6.1, the arguments are similar and theassociated results on the existence of Turing instability are the same. So the detailsare omitted for the discrete case.

7 Conclusion and discussion

The analysis presented here is intended to inform details of the behavior of certainlimiting mathematical systems modeling the TYC strategy for the biological controlof some very specific invasive species. In this work, we study questions that haven’tbeen addressed in (Gutierrez and Teem 2006; Gutierrez 2005; Parshad and Gutierrez2010). Specifically, we study the equilibrium and the stability of model (2.1).We alsoaddress the natural question of how long must the Trojan females be supplied inorder to cause the species to die out. Thus, we investigate what will happen if thisinjection is turned off, after some time. We theoretically study this situation throughbifurcation analysis. We find that the wide-type invasive population can either goextinct or recover. Moreover, extinction and recovery are essentially modulated by aparameter α = βK/(2δ), which is defined by the ratio of the the overall birth ratein terms of the maximal capacity of the ecosystem to two times the per capita deathrate. If α is below the bifurcation value, only extinction can take place; if α is abovethe bifurcation value, both extinction and recovery can happen, and the separatrix ofextinction and recovery is the stable manifold at the saddle. In particular, the basinattraction of the extinction state dramatically shrinks as α increases. Moreover, weidentify a theoretical condition for the eradication strategy to work. Additionally, toaccount for the difficulty of finding females at low levels of the population size, we alsoincorporate 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 havenearly as low population size as that of (2.1) at the optimal termination time, and theremaining sex-reversed supermales are sufficient to eradicate the wild-type females.In addition, we study the possibility of a Turing instability by adding diffusive spatialspread into model (2.1). We find that the inclusion of diffusive spatial spread does notgive rise to a Turing instability, which would have suggested that the TYC eradicationstrategy might be only partially effective, leaving a patchy distribution of the invasivespecies.

We would like to point out that there are a number of interesting questions at thispoint, that would make for interesting future investigations. One could explore theoptimal control strategy, in this context. For instance, assume that the injection rateof trojan fish, μ, is a function of time. We could ask: what’s the optimal controlfor μ(t) such that the eradication strategy works and

∫ T0 μ(t)dt is minimized for a

given amount of time T ? It is also of interest to conduct a bifurcation analysis andconsider the Allee effect and inhomogeneous diffusive spatial spread for the partialdifferential equation model. Here the investigations may have to resort heavily tonumerical methods. Another future direction is to consider stochastic models to studythe probability of extinction of an invasive species in a finite time, since the resultsof the deterministic model in (Parshad and Gutierrez 2010) reveal extinction of aninvasive species is always possible as time goes to infinity.

123


All in all, we hope that our results will be of use in further developing the TYCtheory. The theory, although limited to species with XY sex determination system thatis capable of producing viable progeny during the sex reversal process, is a possiblemeans to combat invasive species of this type.

Acknowledgments This work was supported by the NSF-REU program DMS-0850470. This publicationis based in part on work supported by Award No. KUS-C1-016-04, made by King Abdullah University ofScience and Technology (KAUST).

References

Atkinson K (1992) A survey of numerical methods for solving nonlinear integral equations. J Int Eq Appl4:15–40

Atkinson KE (1997) The numerical solution of integral equations of the second kind. Cambridge Mono-graphs on Applied and Computational Mathematics, Cambridge

Bax NJ, Thresher RE (2009) Ecological, behavioral, and genetic factors influencing the recombinant controlof invasive pests. Ecol Appl 19(4):873–888

Bryan MB, Zalinski D, Filcek KB, Libants S, Li W, Scribner KT (2005) Patterns of invasion and colonizationof the sea lamprey in North America as revealed by microsatellite genotypes. Mol Ecol 14(12):3757–3773

Gutierrez JB (2005) Mathematical analysis of the use of trojan sex chromosomes as means of eradicationof invasive species. Dissertation, Florida State University, Florida

Gutierrez JB, Teem JL (2006) A model describing the effect of sex-reversed YY fish in an establishedwild population: the use of a Trojan Y-Chromosome to cause extinction of an introduced exotic species.J Theor Biol 241(22):333–341

Hamilton WD (1976) Extraordinary sex ratios. A sex-ratio theory for sex linkage and inbreeding has newimplications in cytogenetics and entomology. Science 156(774):477–488

Harmful OTA (1993) Non-indigenous species in the United States. OTA-F-565 U.S. Congress, Office ofTechnology Assessment, Washington

Hill J, Cichra C (2005) Eradication of a reproducing population of Convict Cichlids. Cichlasoma nigrofas-ciatum (Cichlidae) in North-Central Florida. Fla Sci 68:65–74

Howard RD, DeWoody JA, Muir MW (2004) Transgenic male mating advantage provides opportunity forTrojan gene effect in a fish. Proc Natl Acad Sci USA 101:2934–2938

Hurley MA, Matthiesen P, Pickering AD (2004) A model for environmental sex reversal in fish. J TheorBiol 227(2):159–165

Hunter GA, Donaldson EM (1983) Hormonal sex control and its application to fish culture. In: HoarWS, Randall DJ, Donaldson EM (eds) Reproduction: Behavior and fertility control, volume 9B of FishPhysiology. Academic Press, New York, pp 223–303

Lewis MA, Kareiva P (1993) Allee dynamics and the spread of invading organisms. Theor Pop Biol 43:141–158

Katsiadakiand I, Scott AP, Hurstand MR, Matthiessen P, Mayer I (2002) Detection of environmentalandrogens: a novel method based on enzyme-linked immunosorbent assay of spiggin, the stickleback(Gasterosteus aculeatus) glue protein. Environ Toxicol Chem 21(9):1946–1954

Klassen W, Curtis CF (2005) History of the sterile insect technique. In: Dyck VA, Hendrichs J, RobinsonAS (eds) Sterile Insect Technique: Principles and Practice in Area-Wide Integrated Pest Management,Springer-Verlag, Dordrecht, pp 3–36

Knipling EF (1955) Possibilities of insect control or eradication through the use of sexually sterile males.J Econ Entomol 48(4):459–462

Kolmogorov AN, Petrovsky IG, Piskunov NS (1937) Etude de l’équation de la diffusion avec croissancede la quantité de matière et son application un problme biologique. Bulletin Uni- versit dEtat Moscou(Bjul. Moskowskogo Gos. Univ.). Srie Internationale, Section A1, pp 1–26

Myers JH, Simberloff D, Kuris AM, Carey JR (2000) Eradication revisited: dealing with exotic species.Trends Ecol Evol 15:316–320

Nagler JJ, Bouma J, Thorgaard GH, Dauble DD (2001) High incidence of a male-specific genetic markerin phenotypic female Chinook salmon from the Columbia river. Environ Health Perspect 109(1):67–69

123

1756 X. Wang et al.

Parshad RD, Gutierrez JB (2010) On the global attractor of the Trojan Y Chromo—some model. CommunPur Appl Anal 10:339–359

Philips B, Shine R (2004) Adapting to an invasive species: Toxic cane toads induce morphological changein Australian snakes. Proc Natl Acad Sci USA 101(49):17150–17155

Pimentel D, Zuniga R, Morrison D (2005) Update on the environmental and economic costs associatedwith alien-invasive species in the United States. Ecol Econ 52(3):273–288

Odum HT, Allee WC (1954) A note on the stable point of populations showing both intraspecifi cooperationand disoperation. Ecology 35:95–97

Rodda GH, Jarnevich CS, Reed RN (2005) What parts of the US mainland are climatically suitable forinvasive alien pythons spreading from everglades national park? Mol Ecol 14(12):3757–3773

Shafland P, Foote KL (1979) A reproducing population of Serrasalmus humeralis Valenciennes in southernFlorida. Fla Sci 42:206–214

Stephens PA, Sutherlan WJ (1999) Consequences of the Allee effect for behaviour, ecology and conserva-tion. Trends Ecol Evol 14:401–405

Turing A (1952) The chemical basis of morphogenesis. Phil Trans Roc Soc B 237:37–72

123

Analysis of the Trojan Y-Chromosome eradication strategy for an invasive speciesAbstract1 Introduction2 Model description3 Equilibrium analysis of the TYC model3.1 The case μ>03.1.1 Determination of equilibria3.1.2 Stability of equilibria3.1.3 A condition for the eradication strategy

3.2 The case μ=03.2.1 Equilibrium analysis

4 The Allee effect5 The basin of attraction of the extinction state of a wild-type species when μ=05.1 The reduced two-dimensional TYC model5.2 The full TYC model (2.1) when μ=05.3 The reduced 2-dimensional ATYC model

6 Turing instability6.1 A reaction-diffusion model on a line6.2 A stepping-stone type model: reaction and migration in a circle

7 Conclusion and discussionAcknowledgmentsReferences

Date post:	24-Oct-2020
Category:	Documents
Upload:	others
View:	0 times
Download:	0 times

Analysis of the Trojan Y-Chromosome eradication strategy ...mboggess/RESEARCH/trojan.pdfAbstract...

Documents