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On an integro-differential model for pest control in a heterogeneous environment

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Abstract

Insect pests pose a major threat to a balanced ecology as it can threaten local species as well as spread human diseases; thus, making the study of pest control extremely important. In practice, the sterile insect release method (SIRM), where a sterile population is introduced into the wild population with the aim of significantly reducing the growth of the population, has been a popular technique used to control pest invasions. In this work we introduce an integro-differential equation to model the propagation of pests in a heterogeneous environment, where this environment is divided into three regions. In one region SIRM is not used making this environment conducive to propagation of the insects. A second region is the eradication zone where there is an intense release of sterile insects, leading to decay of the population in this region. In the final region we explore two scenarios. In the first case, there is a small release of sterile insects and we prove that if the eradication zone is sufficiently large the pests will not invade. In the second case, when SIRM is not used at all in this region we show that invasions always occur regardless of the size of the eradication zone. Finally, we consider the limiting equation of the integro-differential equation and prove that in this case there is a critical length of the eradication zone which separates propagation from obstruction. Moreover, we provide some upper and lower bound for the critical length.

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References

  • Anaman KA, Atzeni MG, Mayer DG, Stuart MA (1994) Benefit–cost analysis of the use of sterile insect technique to eradicate screwworm fly in the event of an invasion of Australia. Preven Veter Med 20:79–98

    Article  Google Scholar 

  • Atzeni MG, Mayer DG, Butler DG (1992) Sterile insect release method-optimal strategies for eradication of screwworm fly. Math Comput Simul 33:445–450

    Article  Google Scholar 

  • Barbosa P, Castellanos I (2005) Top-down forces in managed and unmanaged habitats. Ecology of predator–prey interactions. Oxford University Press, Oxford

    Google Scholar 

  • Barclay HJ, Mackauer M (1980) The sterile insect release method for pest control: a density dependent model. Environ Entomol 9:810–817

    Article  Google Scholar 

  • Barclay HJ (1981) The sterile release method for population control with interspecific competition. Res Popul Ecol 23:145–155

    Article  Google Scholar 

  • Barclay HJ (1986) Models for pest control: complementary effects of periodic releases of sterile pests and parasitoids. Theor Popul Biol 32:76–89

    Article  Google Scholar 

  • Barclay HJ, Matlock R, Gilchrist S, Suckling DM, Reyes J, Enkerlin W, Vreysen MJB (2011) A conceptual model for assessing the minimum size area for an area-wide integrated pest management program. Intern J Agron 2011:1–12

    Article  Google Scholar 

  • Bates PW, Fife PC, Ren X, Wang X (1997) Traveling waves in a convolution model for phase transitions. Arch Ratl Mech Anal 138:105–136

    Article  MATH  MathSciNet  Google Scholar 

  • Benedict MQ, Robin AS (2003) The first releases of transgenic mosquitoes: an argument for the sterile insect technique. Tren Parasitol 19(8):349–355

    Article  Google Scholar 

  • Berestycki H, Diekmann O, Nagelkerke CJ, Zegeling PA (2009) Can a species keep pace with a shifting climate? Bull Math Biol 71(2):399–429

    Article  MATH  MathSciNet  Google Scholar 

  • Berestycki H, Nadal J-P (2010) Self-organised critical hot spots of criminal activity. Eur J Appl Math 21(4–5):371–399

    Article  MATH  MathSciNet  Google Scholar 

  • Berestycki H, Rodríguez N, Ryzhik L (2013) Traveling wave solutions in a reaction–diffusion model for criminal activity. Multisc Model Simul 11(4):1097–1126

    Article  MATH  Google Scholar 

  • Berestycki H, Rodríguez N (2013) Non-local reaction–diffusion equations with a barrier. Preprint

  • Berryman AA (1967) Mathematical description of the sterile male principle. Can Entomol 99:858–865

    Article  Google Scholar 

  • Bushland RC (1985) Eradication program in the southeastern US. Misc Publ Entomol Soc Am 62:12–15

    Google Scholar 

  • Carr J, Chmaj A (2004) Uniqueness of travelling waves for nonlocal monostable equations. Proc Amer Math Soc 132(8):2433–2439 (electronic)

    Article  MATH  MathSciNet  Google Scholar 

  • Chen X (1997) Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations. Adv Diff Equ 2(1):125–160

    MATH  Google Scholar 

  • Costello WG, Taylor HM (1975) Mathematical models of the sterile male technique of insect control. In: Charnes A, Lynn WR (eds) Mathematical analysis of decision problems in ecology, 5th edn. Springer Verlag, Berlin, pp 8–59

    Google Scholar 

  • Coville J, Dupaigne L (1994) On a nonlocal reaction–diffusion equation arising in population dynamics. Proc Roy Soc Edinb Sect A 137(4):727–755

    Article  MathSciNet  Google Scholar 

  • Coville J, Dupaigne L (2007) On a non-local equation arising in population dynamics. Proc Roy Soc Edinb Sect A 137(4):727–755

    Article  MATH  MathSciNet  Google Scholar 

  • Ermentrout GB, McLeod JB (1993) Existence and uniqueness of travelling waves for a neural network. Proc Roy Soc Edinb 123A:461–478

    Article  MathSciNet  Google Scholar 

  • Esteva L, Mo H (2005) Mathematical model to assess the control of Aedes aegypti mosquitoes by the sterile insect technique. Math Biosci 198(2):132–147

    Article  MATH  MathSciNet  Google Scholar 

  • Fife PC, Peletier LA (1980) Clines induced by variable migration. In: Biological growth and spread (Proceedings Conference, Heidelberg, 1979), vol 38 of Lecture Notes in Biomathematics. Springer, Berlin, pp 276–278

  • Fife PC, Wang X (1998) A convolution model for interfacial motion: the generation and propagation of internal layers in higher space dimensions. Adv Diff Equ 3(1):85–110

    MATH  MathSciNet  Google Scholar 

  • Gatehouse AG (1997) Behavior and ecological genetics of wind-borne migration by insects. Ann Rev Entomol 42:475–502

    Article  Google Scholar 

  • Hendrichs J, Ortiz G, Liedo P, Schwarz A (1982) 6 years of successful medfly program in Mexico and Guatemala. In: Fruit Flies of Economic Importance. CEC/IOBC International Symposium, pp 16–19

  • Hutson V, Martinez S (2003) The evolution of dispersal. J Math Bio 47:483–517

    Article  MATH  MathSciNet  Google Scholar 

  • Hutson V, Grinfeld M (2006) Non-local dispersal and bistability. Eur J Appl Math 17:221–232

    Article  MATH  MathSciNet  Google Scholar 

  • JohnM Kean, SukLing Wee, AndréaEA Stephens (2008) Modelling the effects of inherited sterility for the application of the sterile insect technique. Agricu For Entomol 10(2):101–110

    Article  Google Scholar 

  • Johnson CG (1969) Migration and dispersal of insects by flight. Methuen & Co. Ltd, London

    Google Scholar 

  • Klassen W, Curtis CF (2005) History of the sterile insect technique. In: Dyck et al (eds) Sterile insect technique. Principles and practice in areawide integrated pest management. Springer, Netherlands, pp 3–36

  • Klassen W, Curtis CF (2005) History of the sterile insect technique. Ster Ins Tech, pp 3–36

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

    Article  Google Scholar 

  • Kogan M (ed) (1986) Ecological theory and integrated pest management practice. John Wiley & Sons, New York

    Google Scholar 

  • Kojima KI (1971) Stochastic models for efficient control of insect populations by sterile insect release methods. In sterility principle for insect control or eradication. International Atomic Energy Agency, Vienna

    Google Scholar 

  • Lawson FR (1967) Theory of control of insect populations by sexually sterile males. Ann Entomol Soc Amer 60(4):713–722

    Article  Google Scholar 

  • Lee SS, Baker RE, Gaffney EA, White SA (2013) Optimal barrier zones for stopping the invasion of Aedes aegypti mosquitoes via transgenic or sterile insect techniques. Theor Ecol 6(4):427–442

    Article  Google Scholar 

  • Lewis MA, van den Driessche P (1993) Waves of extinction from sterile insect release. Math Biosci 116:221–245

    Article  MATH  Google Scholar 

  • Lewis TJ, Keener JP (2000) Wave-block in excitable media due to regions of depressed excitability. SIAM J Appl Math 61(1):293–316

    Article  MATH  MathSciNet  Google Scholar 

  • Manoranjan VS, Van Den Driessche P (1986) On a diffusion model for sterile insect release. Math Biosci 79(2):199–208

    Article  MATH  MathSciNet  Google Scholar 

  • Medows ME (1985) Eradication program in the southeastern US. Misc Publ Entomol Soc Am 62:8–11

    Google Scholar 

  • Miller DR, Weidhaas DE (1974) Equilibrium populations during a sterile-male release program. Enviorn Entomol 3(2):211–216

    Article  Google Scholar 

  • Pedigo LP (1998) Entomology and pest management. Prentice-Hall International, Hemel Hempstead

    Google Scholar 

  • Prout T (1978) The joint effects of the release of sterile males and immigration of fertilized females on a density regulated population. Theor Popul Biol 13(1):40–71

    Article  Google Scholar 

  • Serebrovskii AS (1940) On the possibility of a new method for the control of insect pests (in Russian). Zoologicheskii Zhurnal 19:618–630

    Google Scholar 

  • Vanderplank FL (1944) Hybridization between Glossina species and suggested new method for control of certain species of Tsetse. Nature 154(3915):607–608

    Article  Google Scholar 

  • White Steven M, Pejman Rohani (2010) Modelling pulsed releases for sterile insect techniques: fitness costs of sterile and transgenic males and the effects on mosquito dynamics. J Appl Ecol 47(6):1329–1339

    Article  Google Scholar 

Download references

Acknowledgments

This was partially supported by the NSF Postdoctoral Fellowship in Mathematical Sciences DMS-1103765. I am grateful to Ricardo Cortez for bringing this application to my attention. I would like to thank Lenya Ryzhik and Henri Beresticky for their helpful discussions.

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Correspondence to Nancy Rodríguez.

Appendix

Appendix

1.1 Proof of Theorem 3

We state a lemma proved in Coville and Dupaigne (1994) that will enable us to construct a solution from an appropriate super-solution and sub-solution.

Lemma 1

[Coville and Dupaigne (1994)] Let \(f\in C^0(\mathbb R) \cap L^2(\mathbb R).\) There exists a unique solution \(v_0\in C^0(\mathbb R) \cap L^2(\mathbb R)\) to

$$\begin{aligned} \left\{ \begin{array}{l} \mathcal {J}*u - u = f,\\ u(\pm \infty ) = 0. \end{array}\right. \end{aligned}$$
(28)

Proof

(Theorem 3) Let

$$\begin{aligned} f_L(x,u)= \left\{ \begin{array}{l@{\quad }l} f_{ l}(u)&{} x\in (-\infty , 0),\\ f_{ g}(u) &{} x\in [0,L],\\ f_{ r}(u) &{} x\in (0,\infty ),\\ \end{array}\right. \end{aligned}$$
(29)

such that \(f_{ l}(u)\) is monostable with roots zero and one, \(f_{ g}(u)\) is negative and monotonically decreasing with \(f_{ g}(0)=0,\) and \(f{ r}(u)\) is bistable with roots: \(0, a,b\) with \(0<a<b<1\). Refer to Fig. 3 for a visual of the reaction terms \(f_L(x,u)\).

Step 1: (Super-solution and sub-solution) The first step is to construct a super-solution and a sub-solution. To construct a super-solution we let \(g(u)\) be a bistable function such that

$$\begin{aligned} \int \limits _0^1 g(u)\;du = 0, \end{aligned}$$
(30)

and with the additional properties that \(g(u) = f_{ r}(u)\) for \(u\in [0,\delta ],\) \(g(u) = f_{ l}(u)\) for \(u\in [1-\delta ,1],\) for some \(\delta >0\) and \(g(u)\ge f_{ g}(u)\) for all \(u\).

Let \(W(x)\) be the stationary wave solution satisfying

$$\begin{aligned} \left\{ \begin{array}{l} \mathcal {J}*W(x) -W(x)+ g(W(z)) = 0,\\ W(-\infty ) = 1 \;\text {and}\; W(\infty ) = 0. \end{array}\right. \end{aligned}$$

Moreover, we choose the translation of the solution so that \(W(0) = 1-\delta .\) Then, if \(L>0\) is sufficiently large so that \(W(x)<\delta \) for all \(x>L\) and we claim that \(W(x)\) is a super-solution. Indeed, one can verify that

$$\begin{aligned} \mathcal {J}*W-W + f_L(W,x) \le 0. \end{aligned}$$

In the same spirit, let \(g_1(u)\) be a bistable function such that

$$\begin{aligned} \int \limits _0^1 g_1(u)\;du = 0, \end{aligned}$$
(31)

and \(g_1(u) = f_{ g}(u)\) for \(u\in [0,\delta ],\) and \(g_1(u)\le f_{ l} \) for all \(u\). Additionally, if \(h(u):=g(u)+u\) we require

$$\begin{aligned} h'(u)>0. \end{aligned}$$
(32)

Let \(V(x)\) be the stationary traveling wave solution satisfying

$$\begin{aligned} \left\{ \begin{array}{l} \mathcal {J}*V(x) -V(x) + g_1(V(z)) = 0,\\ V(-\infty ) = 1 \;\text {and}\; V(\infty ) = 0, \end{array}\right. \end{aligned}$$

such that \(V(0) = \delta \). Then for all \(L\ge 0\) we obtain that

$$\begin{aligned} \mathcal {J}*V-V + f_L(V,x) \ge 0. \end{aligned}$$

Thus, we have constructed to a super-solution for \(L\) sufficiently large and a sub-solution for all \(L\ge 0\). Note that \(V\in C^2(\mathbb R)\) due to (32) [see Bates et al. (1997)].

Step 2: (Sequence of solutions) We now construct a solution starting with \(V(x)\) and using \(W(x)\) as a barrier. Indeed, let \(u_0(x) = V(x)\) and let \(u_n(x)\) be the solution to

$$\begin{aligned} \left\{ \begin{array}{l} \mathcal {J}*u_n - u_n +\lambda u_n= -f_L(x,u_{n-1}) + \lambda u_{n-1}\\ u_n(-\infty ) = 1\quad u_n(\infty ) = 0. \end{array}\right. \end{aligned}$$

We first show that such \(u_n(x)\) exists. Following Coville and Dupaigne (1994) we choose a \(\phi \in C_c^\infty \) satisfying \(\Vert \phi \Vert _{L^1} =1\) and define

$$\begin{aligned} \varphi (x) = \int \limits _x^\infty \phi (s)\;ds. \end{aligned}$$

Then, \(v_n=u_n-\varphi \) satisfies

$$\begin{aligned} \left\{ \begin{array}{l} \mathcal {J}*v_n - v_n + \lambda v_n= -\mathcal {J}*\varphi + \varphi - f_L(x,v_{n-1}+\varphi ) + \lambda v_{n-1} \\ v_n(\pm \infty ) = 0. \end{array}\right. \end{aligned}$$

Note, that by our choice of \(u_0\) we know that \(v_0\in C^0\cap L^2(\mathbb R)\) because \(\varphi \equiv 0\) for \(x\) large and \(\varphi \equiv 1\) for \(x\) negative enough. Note that to apply Lemma 1 we need \(f_L(x,v_{0}+\varphi )\in C^0(\mathbb R).\) Since the reaction term \(f_L(x,u)\) is discontinuous we regularize it by convolving with the standard mollifier and denote it by \(f^{\epsilon _1}_L(x,v_{0}+\varphi ),\) with \(\epsilon _1\) being the regularizing parameter. We first find a solution to

$$\begin{aligned} \left\{ \begin{array}{l} \mathcal {J}*v_1 - v_1 + \lambda v_1= -\mathcal {J}*\varphi + \varphi - f^{\epsilon _1}_L(x,v_{0}+\varphi ) + \lambda v_{0}. \\ v_n(\pm \infty ) = 0. \end{array}\right. \end{aligned}$$
(33)

Now, for \(x<0\) we have, using the mean value theorem and \(f^{\epsilon _1}_l(1)=0\), that

$$\begin{aligned} \left| f^{\epsilon _1}_L(x,v_{0}+\varphi )\right| < \Vert \partial _x f^{\epsilon _1}_l\Vert _\infty \left| v_0+\varphi -1\right| = \Vert \partial _x f^{\epsilon _1}_l\Vert _\infty \left| V-1\right| \in L^2(\mathbb R^-) \end{aligned}$$

and for \(x>L\) we have that \(f^{\epsilon _1}_r(0)=0\)

$$\begin{aligned} \left| f^{\epsilon _1}_L(x,v_{0}+\varphi )\right| < \left| v_0+\varphi \right| = \Vert \partial _x f^{\epsilon _1}_r\Vert _\infty \left| V\right| \in L^2(\mathbb R^+). \end{aligned}$$

Thus, \(f^{\epsilon _1}_L(x,v_0+\varphi )\in L^2(\mathbb R)\). Finally, the fact that \(\mathcal {J}*\varphi - \varphi \in L^2(\mathbb R)\) follows from Lemma 3.1 in Coville and Dupaigne (1994). Now, applying Lemma 1 gives a solution to (33). However, since the estimates above are independent of the regularizing parameter we can take the limit as \(\epsilon _1\rightarrow \infty \) to obtain a solution to the original problem. We omit the details as the procedure is standard. Repeating this process iteratively gives us a sequence of solutions, \(\left\{ u_n\right\} _{n\in \mathbb {N}}\).

Step 3: (Passing to the limit as \(n\rightarrow \infty \)) From induction and the maximum principle we know that

$$\begin{aligned} V(x)\le u_n(x)\le U(x) \end{aligned}$$

for all \(n\in \mathbb {N}\). Furthermore, if we define \(w_n = u_n(x-\tau )-u_n(x)\) for some \(\tau > 0\) then \(w_n\) solves

$$\begin{aligned} \mathcal {J}*w_n - w_n + \lambda w_n&= -f_L\left( x,u_{n-1}(x-\tau )\right) +\lambda u_{n-1}(x-\tau ) \\&+ f_L\left( x,u_{n-1}(x)\right) -\lambda u_{n-1}(x). \end{aligned}$$

Now, we choose \(\lambda >\Vert f'(u)\Vert \) so that \(-f_L+\lambda \) is non-decreasing and positive; thus, making the right hand side of the above equality non-negative for \(u_1\) given that \(V\) is monotone decreasing. An application of the maximum principle implies that \(w_1\) is always non-negative. Iterating this idea gives that \(u_n(x)\) is non-increasing in \(x\). Invoking, Helly’s lemma we obtain that a subsequence converges to a non-increasing solution \(u(x)\) also satisfying \(V(x)\le u(x)\le W(x).\) Moreover, the dominated convergences implies

$$\begin{aligned} \mathcal {J}*u_n - u_n \rightarrow \mathcal {J}*u - u. \end{aligned}$$

Note also that \(\mathcal {J}*u_n \in C^\infty (\mathbb R)\) and

$$\begin{aligned} \mathcal {J}*u_n = u_n - f_L(x,u_n), \end{aligned}$$

which implies that \(u_n - f_L(x,u_n)\in C^\infty (\mathbb R).\) This gives uniform convergence in \(u_n-f_L(x,u_n)\rightarrow u-f_L(x,u).\) This concludes the existence. We are left to show that the solution must be discontinuous.

Step 4: (Discontinuity) The solution \(u(x)\) satisfies

$$\begin{aligned} \mathcal {J}*u = u+f_L(x,u), \end{aligned}$$

and since \(\mathcal {J}*u\) is smooth then \(u+f_L(x,u)\) must be smooth. This implies that

$$\begin{aligned}&\lim _{x\rightarrow 0^-}u(x) + f_{ l} (u(x)) = \lim _{x\rightarrow 0^+}u(x) + f_{ g} (u(x))\\&\lim _{x\rightarrow 0^-}u(x) + f_{ g} (u(x)) = \lim _{x\rightarrow 0^+}u(x) + f_{ r} (u(x)). \end{aligned}$$

Since these functions do not intersect we can denote \(u(0^+)\) to be \(\lim _{x\rightarrow 0^+} u(x)\) (similarly for the other limits) then we have that

$$\begin{aligned} u(0^-) + f_{ l} (u(0^-))&= u(0^+) + f_{ g} (u(0^+))\\ u(0^-) + f_{ g} (u(0^-))&= u(0^+) + f_{ r} (u(0^+)). \end{aligned}$$

With this we conclude the proof. \(\square \)

1.2 Proof of Theorem 6

Proof

Let \(u(x)\) be the solution to (12) with reaction term \(f_L(x,u)\) defined in the hypotheses of the theorem. Since \(s_r=0\) in (11) then \(f_{ r}(x)\) is monostable, which implies that \(f(s)\ge 0\) for \(s\in (0,1)\). For \(x>L\) we have that

$$\begin{aligned} \mathcal {J}*u - u = -f_{ r}(u) \le 0, \end{aligned}$$
(34)

with equality only if \(u\equiv 1\). We know that there is a \(\delta >0\) such that \(u(L)=\delta \); hence, we know that \(u(x)\ge \delta \) for \(x>L\). We claim that the limit of the reaction term is zero,

$$\begin{aligned} \lim _{x\rightarrow \infty }f(u(x)) = 0, \end{aligned}$$

which implies that \(\lim _{x\rightarrow \infty } u(x) = 1\) and from this we can conclude. To prove this claim we let \(\mathcal {J}= \mathcal {J}_\epsilon \) satisfy \(H0-H2\) with \(\epsilon <<1.\) We will approximate the non-local operator by the Laplacian with appropriate diffusion coefficient; indeed (provided \(u\) is sufficiently smooth),

$$\begin{aligned} \mathcal {J}_\epsilon * u - u \approx \epsilon ^2u_{xx} + \mathcal {O}(\epsilon ^2). \end{aligned}$$
(35)

Let \(v\) be the solution to

$$\begin{aligned}&-\epsilon ^2 v_{xx} = 1\quad \text {on}\;B(0,1)\\&\quad v = 0 \quad \;\text {on} \;\partial B(0,1). \end{aligned}$$

In fact, \(v(x) = \frac{1-x^2}{2\epsilon ^2}\) is the solution with a maximum value at \(x=0\) and equal to \(\frac{1}{2\epsilon ^2}.\) We define

$$\begin{aligned} \varGamma (y) =\min \left\{ f(s): s\in [\delta , u(y)]\right\} \end{aligned}$$
(36)

such that \(u(L)=\delta \) and claim that

$$\begin{aligned} \varGamma (y)\le \frac{2}{(y-L)^2}. \end{aligned}$$
(37)

Note that if (37) is holds then we are done by the definition of \(\varGamma \). Assume, to get a contradiction, that (37) does not hold, so that there exists an \(y_0>L\) such that

$$\begin{aligned} \varGamma (y_0)> \frac{2}{(y_0-L)^2}. \end{aligned}$$

Fix \(R\) such that \(L<R<y_0\) sufficiently close to \(y_0\) so that

$$\begin{aligned} \frac{\varGamma (y_0)}{2}> \frac{1}{(R-L)^2}. \end{aligned}$$

Note that \(y_0\) cannot be a minimum of \(u(x)\) by (34). Thus, there exists a \(R<y_1<y_0\) so that \(\delta <u(y_1)<u(y_0)\). Let \(r=R-L\) and define

$$\begin{aligned} z(y) = \varGamma (y_0)r^2 v\left( \frac{y-y_1}{r}\right) , \end{aligned}$$

which satisfies

$$\begin{aligned}&-\epsilon ^2 z_{xx} = \varGamma (y_0)\; \text {on}\;B(y_1,r)\\&\quad z = 0 \quad \;\text {on} \;\partial B(y_1,r). \end{aligned}$$

Note that

$$\begin{aligned} \max _{x\in B(y_1,r)}z(x) =\frac{\varGamma (y_0)r^2}{2\epsilon ^2}. \end{aligned}$$

Now, consider \(z^\tau (x) := \tau z(x)\) for \(\tau >0\). For \(\tau \) small enough we have that

$$\begin{aligned} z^\tau <\delta <u(x) \; x\in B(y_1,r). \end{aligned}$$

Let \(\tau ^o\) be the first value where the graph of \(z^{\tau _o}\) touches the graph of \(u(x)\) in \(B(y_1,r)\). Then there exists an \(x_0\) in the interior of \(B(y_1, r)\) (as \(z=0\) on the boundary) where \(\tau ^oz(x_0) = u(x_0)\) and \(z^{\tau ^0}(x)\le u(x)\) for \(x\in B(y_1, r).\) So we have

$$\begin{aligned} u(x_0)\le \tau ^0z(x_0) \le \tau ^0z(y_1) = \frac{\tau ^o\varGamma (y_0)r^2}{2\epsilon ^2}\le u(y_1) <u(y_0)<1. \end{aligned}$$

From this follows that

$$\begin{aligned} \tau ^0 < \frac{2\epsilon ^2}{\varGamma (y_0)r^2}<1, \end{aligned}$$

by the choice of \(R\) and because \(\epsilon <<1\). Now, we define \(w:=\tau ^oz - u\) and we know that \(w\le 0\) for \(x\in B(y_1,r)\) and \(w(x_0) = 0\). Moreover, we know that \(u(x_0) <u(y_0)\) so there exists a neighborhood centered at \(x_0\), \(\mathcal {N}(x_0)\) such that

$$\begin{aligned} u(x)<u(y_0) \quad \text {for}\;x\in \mathcal {N}(x_0). \end{aligned}$$

Thus,

$$\begin{aligned} \mathcal {J}*u - u \le -\varGamma (y_0)\quad \text {for}\;x\in \mathcal {N}(x_0), \end{aligned}$$

and by (35) provided \(\epsilon \) is sufficiently small

$$\begin{aligned} \epsilon ^2 u_{xx} < -\varGamma (y_0) \end{aligned}$$

and

$$\begin{aligned} \epsilon ^2 w_{xx} \ge -\tau ^0\varGamma (y_0) +\varGamma (y_0)> 0, \end{aligned}$$

for \(x\in \mathcal {N}(x_0)\). This contradicts the fact that \(w(x)\) has a local minimum at \(x=x_0\) and we conclude that \(f(u)\rightarrow 0\) as \(x\rightarrow \infty \) which implies that \(u\rightarrow 1\) as \(x\rightarrow \infty \) in the case when \(\epsilon \) is small. For general \(\epsilon \) we know that the solution is simply a rescaling of the problem in \(\epsilon \). Indeed, if \(u(x,t)\) is a solution to

$$\begin{aligned} u_t = \mathcal {J}_1*u - u +f_L(x,u), \end{aligned}$$

then \(v(x):=u(\epsilon x)\) is a solution to

$$\begin{aligned} v_t = \mathcal {J}_\epsilon *v - v+f_{\epsilon L}(x,v). \end{aligned}$$

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Rodríguez, N. On an integro-differential model for pest control in a heterogeneous environment. J. Math. Biol. 70, 1177–1206 (2015). https://doi.org/10.1007/s00285-014-0793-8

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