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1 SIAM J MATH ANAL Vol 46, No, pp c 4 Society for Industrial and Applied Mathematics Downloaded //4 to Redistribution subject to SIAM license or copyright; see EXISTENCE OF SYMMETRIC AND ASYMMETRIC SPIKES FOR A CRIME HOTSPOT MODEL HENRI BERESTYCKI, JUNCHENG WEI, AND MATTHIAS WINTER Abstract We study a crime hotspot model suggested by Short, Bertozzi, and Brantingham in [SIAM J Appl Dyn Syst, 9, pp ] The aim of this work is to establish rigorously the formation of hotspots in this model representing concentrations of criminal activity More precisely, for the one-dimensional system, we rigorously prove the existence of steady states with multiple spikes of the following types: i multiple spikes of arbitrary number having the same amplitude symmetric spikes, and ii multiple spikes having different amplitude for the case of one large and one small spike asymmetric spikes We use an approach based on Lyapunov Schmidt reduction and extend it to the quasilinear crime hotspot model Some novel results that allow us to carry out the Lyapunov Schmidt reduction are i approximation of the quasilinear crime hotspot system on the large scale by the semilinear Schnakenberg model, and ii estimate of the spatial dependence of the second component on the small scale which is dominated by the quasilinear part of the system The paper concludes with an extension to the anisotropic case Key words crime model, reaction-diffusion systems, multiple spikes, symmetric and asymmetric, quasilinear chemotaxis system, Schnakenberg model, Lyapunov Schmidt reduction AMS subject classifications Primary, 35J5, 35B45; Secondary, 36J47, 9D5 DOI 37/39744 Introduction: The statement of the problem Pattern forming reactiondiffusion systems have been and are applied to many phenomena in the natural sciences Recent works have also started to use such systems to describe macroscopic social phenomena In this direction, Short, Bertozzi, and Brantingham [9] have proposed a system of nonlinear parabolic partial differential equations to describe the formation of hotspots of criminal activity Their equations are derived from an agentbased lattice model that incorporates the movement of criminals and a given scalar field representing the attractiveness of crime The system in one dimension reads as follows: A t = A xx A + ρa + A x in, L, ρ t = D ρ x ρ A A x ρa + γx in, L x Here A is the attractiveness of crime and ρ denotes the density of criminals The rate at which crimes occur is given by ρa When this rate increases, the number of Received by the editors May 9, 3; accepted for publication in revised form November 7, 3; published electronically February 6, 4 École des Hautes Études en Sciences Sociales, CAMS, 7544 Paris cedex 3, France hb@ehessfr The research of this author was supported by the European Research Council under the European Union s Seventh Framework Programme FP/7-3 / ERC Grant Agreement ReaDi - Reaction-Diffusion Equations, Propagation, and Modelling, and by NSF FRG grant DMS-65979, Emerging Issues in the Sciences Involving Non-standard Diffusion Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong, China, and Department of Mathematics, University of British Columbia, Vancouver, BC, V6T Z, Canada wei@mathcuhkeduhk The research of this author was supported by a GRF grant from RGC of Hong Kong and an NSERC grant from Canada Department of Mathematical Sciences, Brunel University, Uxbridge UB8 3PH, United Kingdom matthiaswinter@brunelacuk 69

2 69 HENRI BERESTYCKI, JUNCHENG WEI, AND MATTHIAS WINTER Downloaded //4 to Redistribution subject to SIAM license or copyright; see criminals is reduced while the attractiveness increases The second feature is related to the well-documented occurrence of repeat offenses The positive function A x is the intrinsic static attractiveness which is stationary in time but possibly variable in space The positive function γx is the source term representing the introduction rate of offenders per unit area For the precise meanings of the functions A x and γx, we refer the reader to [9,, ] and the references therein This paper is concerned with the mathematical analysis of the one-dimensional version of this system Let us describe our approach Setting the system is transformed into v = ρ A, A t = A xx A + va 3 + A x in, L, A v t = D A v x x va3 + γx in, L We always consider Neumann boundary conditions A x =A x L =ρ x =ρ x L =v x =v x L = Note that v is well-defined and positive if A and ρ are both positive The parameter < represents nearest neighbor interactions in a lattice model for the attractiveness We assume that it is very small, which corresponds to the temporal dependence of attractiveness dominating its spatial dependence This models the case of attractiveness propagating rather slowly, ie, much slower than individual criminals It is a realistic assumption if the criminal spatial profile remains largely unchanged, or, in other words, if the relative crime-intensity does only change very slowly This appears to be a reasonable assumption since it typically takes decades for dangerous neighborhoods, ie, those attracting criminals, to evolve into safe ones, and vice versa Roughly speaking, a k-spike solution A, v to is such that the component A has exactly k local maximum points In this paper, we address the issue of existence of steady states with multiple spikes in the following two cases: symmetric spikes same amplitudes or asymmetric spikes different amplitudes Our approach is by rigorous nonlinear analysis We apply Lyapunov Schmidt reduction to this quasilinear system In this approach, to establish the existence of spikes, we derive the following new results: i approximation of the crime hotspot system on the large scale of order one by the semilinear Schnakenberg model see section 3, in particular 33; ii estimate of the spatial dependence of the second component on the small scale of order, dominated by the quasilinear part of the system see section 6, in particular inequalities 6 63 We remark that asymmetric multiple spike steady states of k small and k large spikes are an intermediate state between two different symmetric multiple spike steady states of k + k spikes for which all spikes are fully developed and k spikes for which the small spikes are gone These rigorous results shed light on the formation of hotspots for the idealized model of criminal activity introduced in [9]

3 EXISTENCE OF SPIKES FOR A CRIME HOTSPOT MODEL 693 Downloaded //4 to Redistribution subject to SIAM license or copyright; see Let us now comment on previous works As far as we know, there are three mathematical works related to the crime model Short, Bertozzi, and Brantingham [9] proposed this model based on mean field considerations They have also performed a weakly nonlinear analysis on about the constant solution γ A, ρ = γ + A, γ + A assuming that both A x andγx are homogeneous Rodriguez and Bertozzi have further shown local existence and uniqueness of solutions [7] In [5], Cantrell, Cosner, and Manásevich have given a rigorous proof of the bifurcations from this constant steady state On the other hand, in the isotropic case, Kolokolnikov, Ward, and Wei [3] have studied existence and stability of multiple symmetric and asymmetric spikes for using formal matched asymptotics They derived qualitative results on competition instabilities and Hopf bifurcation and gave some extensions to two-space dimensions The present paper provides rigorous justification for many of the results in [3] and also derives some extensions In particular, we establish here the following three new results: First, we reduce the quasilinear chemotaxis problems to a Schnakenbergtype reaction-diffusion system and prove the existence of symmetric k spikes Second, this paper gives the first rigorous proof of the existence of asymmetric spikes in the isotropic case Third, we study the pinning effect in an inhomogeneous setting A x and γx The stability of these spikes is an interesting issue which should be addressed in the future We should mention that another model of criminality has been proposed and analyzed by Berestycki and Nadal [] In a forthcoming paper [3], we shall study the existence and stability of hotspots spikes in this system as well It is quite interesting to observe that both models admit hotspot spike solutions The structure of this paper is as follows We formally construct a one-spike solution in section in which we state our main results In section 3 we show how to approximate the crime hotspot model by the Schnakenberg model Section 4 is devoted to the computation of the amplitudes and positions of the spikes to leading order Nondegeneracy conditions are derived in section 5 These are required for the existence proof, given in sections 6 8 In section 6 we introduce and study the approximate solutions In section 7 we apply Lyapunov Schmidt reduction to this problem Last, we solve the reduced problem in section 8 and conclude the existence proof In section 9 we extend the proof of single spike solution to the case when both A x andγx are allowed to be inhomogeneous Finally, in section we discuss our results and their significance and mention possible future work and open problems Steady state: Formal argument for leading order and main results Before stating the main results, we first construct a time-independent spike on the interval [, L] located at some point x The construction here is carried out using classical matched asymptotic expansions In the inner region, we assume that v is a constant v in leading order: vx v, Then, if <, the equation for A becomes x x A A + v A 3 + A x =

4 694 HENRI BERESTYCKI, JUNCHENG WEI, AND MATTHIAS WINTER Rescaling Downloaded //4 to Redistribution subject to SIAM license or copyright; see we get Ax =v / Ây, y = x x,  yy  + Â3 + A x + yv / = We assume that v as Then, at leading order, Ây wy, where w is the unique even solution of the ODE so that so that In the outer region, we assume that w yy w + w 3 = wy = sechy va 3, A A x x We also assume that D = ˆD,where ˆD is a positive constant, and we estimate L va 3 dx v / w 3 dy Integrating the second equation in, we then have L v / w 3 dy γxdx, v w3 dy L γxdx We remark that w 3 dy = wdy = π so that In particular, we obtain A x+ 3 Ax π v L γx L γxdx π w x x, x = O, A x, x O

5 EXISTENCE OF SPIKES FOR A CRIME HOTSPOT MODEL 695 Downloaded //4 to Redistribution subject to SIAM license or copyright; see Now we state our main theorems on the existence of multispike steady states for system We discuss two cases In the case of isotropic coefficients A x Constant, γx Constant, we will consider two types of solutions: i multiple spikes of arbitrary number having the same amplitude symmetric spikes; ii multiple spikes having different amplitude for the case of one large and one small spike asymmetric spikes In the case of anisotropic coefficients A x andγx, we will consider the existence of a single-spike solution Our first result concerns the existence of multiple spikes of arbitrary number having the same amplitude symmetric spikes Theorem Assume that D = ˆD for some fixed ˆD > and 4 A x A,γx Ā A, where Ā>A Then, provided > is small enough, problem has a K-spike steady state A,v which satisfies the following properties: 5 A x =A + j= v j w x t j 6 v t i = v i, i =,,K, where 7 t i t i, i =,,K, with 8 t i = i K K and 9 v i = v i with v i = π K L, i =,,K, + O log, +O log, i =,,K, Ā A L, i =,,K Remark Note that in 5 a two-term expansion of the solution A is given, where for each spike the term v w x t j j ofordero is the leading term in the inner solution and the term A of order O is the leading term in the outer solution By using the operator T [Â] defined in 3 this two-term expansion carries over to ˆv as well The same remark applies to 3 and 5 The two-term expansion agrees with that in [3] The next result is about asymmetric two-spikes

6 696 HENRI BERESTYCKI, JUNCHENG WEI, AND MATTHIAS WINTER Downloaded //4 to Redistribution subject to SIAM license or copyright; see Theorem 3 Under the same assumption as in Theorem, withd = ˆD some fixed ˆD >, and supposing moreover that and π ˆDA /4 Ā A 3/4 L π ˆDA /4 Ā A 3/4 L, 5 then, for > small enough, problem has an asymmetric -spike steady state A,v which satisfies the following properties: 3 A x =A + j= x t w i + O log, v i 4 v t i = vi, i =,,K, where t i and v i satisfy 7 and 9, respectively The limiting amplitudes v i and positions t i are given as solutions of 4 and 46 Condition is a kind of nondegeneracy condition Note that in the case of asymmetric spikes we explicitly characterize the points of nondegeneracy The last theorem is about the existence of a single-spike solution in the anisotropic case Theorem 4 Assume that > is small enough and D = ˆD for some fixed ˆD > Then problem has a single-spike steady state A,v which satisfies the following properties: 5 A x =A x+ x t w + O log, v 6 v t = v, where 7 t t, and 8 v = t π L γxdx γxdx = L t γxdx +O log We notice that in the anisotropic case, the single-spike location is determined only by the function γtdt and A x has no effect at all Note also that the location t is uniquely determined by the condition t 9 γxdx = γxdx With more computations, it is possible to construct multiple asymmetric spikes in the isotropic case, and also multiple spikes in the anisotropic case Since the statements and computations are complicated, we will not present them here We refer the reader to [6] for some results in this direction L for

7 EXISTENCE OF SPIKES FOR A CRIME HOTSPOT MODEL 697 Downloaded //4 to Redistribution subject to SIAM license or copyright; see 3 Scaling and approximation by the Schnakenberg model We will use the following notation for the domain and the rescaled domain, respectively: 3 Ω =, L, Ω = L, L This section is devoted to the reduction of the system to a particular Schnakenberg-type reaction-diffusion equation in which no chemotaxis appears Based on the computations in section, we rescale the solution and the second diffusion coefficient as follows: 3 A = A x+ Â, v = ˆv, D = ˆD Then the steady-state problem becomes =  xx  +ˆvA + Â3 + 3 A, x Ω, = ˆD A x+  ˆv x ˆvAx+Â3 + γx, x Ω x We will consider the case when and ˆD is constant, with Neumann boundary conditions A key observation of this paper is that the solutions of problem 3 are very close to the solutions of the Schnakenberg model, 33 =  xx  +ˆvA + Â3 + 3 A, x Ω, = ˆD A ˆv x x ˆvA + Â3 + γx, x Ω, with Neumann boundary conditions To see this, we first consider the following linear problem: ˆDaxv x x = fx, <x<l, 34 v x =v x L =, where a C, L, ax c> for all x, L andf L, L We compute So axv x x = 35 v x x = ax and hence 36 vx v = ft dt ˆD ft dt ˆD s ftdt ds, ˆDas

8 698 HENRI BERESTYCKI, JUNCHENG WEI, AND MATTHIAS WINTER Downloaded //4 to Redistribution subject to SIAM license or copyright; see which can be rewritten as 37 vx v = ˆD where K a x, s = s K a x, sfs ds, at dt Remark 5 We note that the kernel K a x, s is an even odd function if ax is an odd even function More precisely, if ax =±a x, then 38 K a x, s = K a x, s Remark 6 We note that v is an even odd function if f is even and a is even odd More precisely, using we compute Integration yields and ft dt = L ft dt = v x x = ax vx v = = ˆD ˆDas ft dt ˆD s K a x, sfs ds x iff is even, ft dt ds v x v = ˆD K a x, sfs ds = ˆD K a x, sf s ds = ± ˆD K a x, sfs ds = ±vx v if a is an even odd function using 38 Similarly, if f is odd and a is odd even, then v is an even odd function Integrating 34, we derive the necessary condition 39 L fx dx = Note that on the other hand v defined by 34 satisfies the boundary conditions v x =v x L =, provided that 39 holds This follows from 35

9 EXISTENCE OF SPIKES FOR A CRIME HOTSPOT MODEL 699 Let us now consider ax = A + γ w x,wherew>andwy e y as y Then we claim that Downloaded //4 to Redistribution subject to SIAM license or copyright; see 3 K a x, s =K A x, s+o s x +O [s, x] log, log Note that 3 is an L estimate for K a x, s In fact, we have x x [ s A + dt = w s A dx + s A + ] w A dt, where [ ] s A A + w = dy+ [s/,x/] { y >log } dt = s A w + w A + w [s/,x/] { y <log } dy dy The first term is O x s sincew = O andso Aw+w A +w = O For the second term, observing that Aw+w A +w = O we derive 3 All these estimates are in the L norm Thus, v satisfies 3 vx v = ˆD K A x, sfsds + O x s +log fs ds Remark 7 The estimates 3 and 3 also hold if ax = A + γ w x x + φ, where φx satisfies φx Cmaxe x x, This is the class of functions that we will work with This is also the motivation for our choice of the norm defined in 74 Therefore, we can approximate steady states for the crime hotspot model by the Schnakenberg model as follows: Given Â>, let ˆv = T [Â] be the unique solution of the following linear problem: 3 ˆD A + ˆv x ˆvA + x Â3 + γx =, ˆv x =ˆv x L = <x<l, Then, by the maximum principle, the solution T [Â] is positive

10 7 HENRI BERESTYCKI, JUNCHENG WEI, AND MATTHIAS WINTER Downloaded //4 to Redistribution subject to SIAM license or copyright; see By the previous computations and remarks, if  = w + φ with φ Cmax e x x,, it follows that 33 T [Â] =v + O log in H, L, where v satisfies ˆD A 34 v x x v A + Â3 + γx =, vx =vxl = <x<l, We adapt an approach based on Lyapunov Schmidt reduction which has been applied to the semilinear Schnakenberg model in [] and extend it to the quasilinear crime hotspot model This method has also been used to study spikes for the one-dimensional Gierer Meinhardt system in [5, 6] as well as the two-dimensional Schnakenberg model in [7] We refer the reader to the survey paper [4] and the book [8] for references Multiple asymmetric spikes for the one-dimensional Schnakenberg model have been considered using matched asymptotics in [] Existence and stability of localized patterns for the crime hotspot model have been studied by matched asymptotics in [3], and results on competition instabilities and Hopf bifurcation have been shown, including some extensions to two space dimensions We remark that another approach for studying multiple spikes in one-dimensional reaction-diffusion systems is the geometric singular perturbation theory in dynamical systems For results and methods in this direction we refer the reader to [7, 8] and the references therein 4 Computation of the amplitudes and positions of the spikes In this section, we study 3 in the isotropic case 4 In particular, we compute the amplitudes and positions to leading order We consider symmetric multispike solutions with any number of spikes and asymmetric multispike solutions with one small and one large spike We first write down the system for the amplitudes in case of a general number K of spikes, where we have either K spikes of the same amplitude or k small and k large spikes with k + k = K We will first solve this system in the case of symmetric spikes Then we will choose k = k = and solve this system in this special case of asymmetric spikes Integrating the right-hand side of the second equation in 3, we compute for v j = lim ˆv t j 4 π =Ā A L vj j= Solving the second equation in 3, using 36 in combination with the approximation 33, we get s ˆv x ˆv = ˆDA +  ˆv A +  3 Ā + A dt ds s = ˆDA ˆv A +  3 Ā + A dt ds + O log,

11 Downloaded //4 to Redistribution subject to SIAM license or copyright; see where ˆDA = ˆDA EXISTENCE OF SPIKES FOR A CRIME HOTSPOT MODEL 7 i π Hs t ˆv t j j π j= j= i π x t ˆv t j j j= Hx = j= ˆv t j ˆv t j π { if x, if x< s + L L x + L 4L Taking the limit and setting x = t i = lim t i, we derive 4 v i = ˆDA = ˆDA ds + O + O i π t i t j π t i + L + C vj vj 4L j= j= π vj t i t j π vj 4L t i + C j= j= log log, for some real constants C,C independent of i, where the last identity in 4 uses 44, which we now explain We use an assumption on the position of spikes that can be stated as follows: 43 i π F i t,t,,t K := π + π t i + L =, i =,,K vi vj vj L j= Note that 43 will be derived later on in section 8 see 86 We rewrite 43 and compute 44 = i j= v j i + v i j= j=i+ From 4 and 44, we derive 45 v i+ v i = j= vj v j vj j= vj j= t i + L L t i L = π t ˆDA i+ t i vi vi+ In the next two subsections we solve these equations for the amplitudes of the spikes in the cases of both symmetric and asymmetric spikes

12 7 HENRI BERESTYCKI, JUNCHENG WEI, AND MATTHIAS WINTER Downloaded //4 to Redistribution subject to SIAM license or copyright; see 4 Symmetric spikes We first consider the case of symmetric spikes, where v i = v is independent of i =,,K, and compute the amplitude v and the positions t i From 4, we get π K v = Ā A L From 45, the positions are t i = + i K L, i =,,K The proof of the existence of multiple symmetric spikes follows from the construction of a single spike in the interval L K, L K The proof of the existence of a single spike uses the implicit function theorem in the space of even functions, for which the Lyapunov Schmidt reduction method is not needed This proof can easily be obtained by specializing the proof given for multiple asymmetric spikes given below In this case, the proof can thus be simplified Therefore we omit the details 4 Asymmetric spikes Combining 44 and 45, we get π L v i v i+ = ˆDA K j= vj This implies that there are only two different amplitudes, which we denote by v s v l, appearing k and k times, respectively Hence we get 46 v s v l = where get Multiplying 44 by ti v i C = π ˆDA π L ˆDA vs k + k vl and subtracting 4 from the result, we get k t j sgnt j t i + k π vj ˆDA vj j= + if α>, sgnα = if α =, if α< j= t i L, Next we determine v s,v l from 4 and 46 Substituting 4 into 46, we v l = v s π ˆDA 4 Ā A Plugging this equation into 4 gives C z + =, z where π ˆDA C = /4 vs Ā A k k 3/4, z = ˆDA /4 Ā A /4 k L π k

13 EXISTENCE OF SPIKES FOR A CRIME HOTSPOT MODEL 73 To determine a solution, we need to satisfy the necessary condition C <, which can be summarized as Downloaded //4 to Redistribution subject to SIAM license or copyright; see π ˆDA /4 Ā A k k 3/4 < L The second necessary condition is given by v s <v l, which is equivalent to z< This implies the following cases Case i: k k If C k k + k then there exists exactly one solution with z On the other hand, if C k k + k k k <, k k >, k then there exists no solution with z< k Case ii: k >k IfC>, there is no solution If C <and C k k + k k <, k then there exists exactly one solution with z k If C <and k k C + >, k k k k k then there exist exactly two solutions with z< k Special case: k = k = Finally, we consider the special case k = k = which belongs to Case i in the previous classification, and we have the following results: If C <, then there is one solution with z< If C >, then there is no solution with z> 5 Existence and nondegeneracy conditions We now describe a general scheme of Lyapunov Schmidt reduction We refer the reader to the survey paper [4] for more details Essentially this method divides the problem of solving nonlinear elliptic equations and systems into two steps In the first step, the problem is solved up to multipliers of approximate kernels In the second step one solves algebraic equations in terms of finding zeros of the multipliers In this section, we linearize 3 around the approximate solution and derive the linearized operator as well as its nondegeneracy conditions, ie, conditions such that the resulting linear operator is uniformly invertible

14 74 HENRI BERESTYCKI, JUNCHENG WEI, AND MATTHIAS WINTER Linearizing 3 around the solution, we get Downloaded //4 to Redistribution subject to SIAM license or copyright; see 5 = φ xx φ +3ˆvA + Â φ + ψa + Â3 in Ω, = ˆD A + Â ψ x 3 ˆvA + Â φ ψa + Â3 with Neumann boundary conditions x + ˆD A + Â φˆv x x in Ω φ x =φ x L =ψ x =ψ x L = Note that for the second equation of 5 we have the necessary condition L 3 ˆvA + Â φ ψa + Â3 dx =, which follows by integrating the equation and using the Neumann boundary conditions for ψ and v In the limit weget [ ] π ψ j +3 w φ j= v 3/ j dy =, j where ψ j = ψx j The second equation of 5 can be solved as follows, using formula 36 and estimate 33: ψx ψ = s s ˆDA + Â = ˆDA Note that the contributions from the term [ ˆD A + Â φˆv x x + 3 ˆvA + Â φ + ψa + Â3 ]dt ds 3 ˆvA + Â φ + ψa + Â3 dt ds + O log ˆD A + Â φˆv x can be estimated by O log since φ vanishes in the outer expansion In the limit, we get i ψx ψ = π i ψ ˆDA j Hs t j= v 3/ j +3 w φ j dyhs t j ds j j= = i π i 5 ψ ˆDA j x t j= v 3/ j +3 w φ j dy x t j, j j= x where t i <x t i

15 EXISTENCE OF SPIKES FOR A CRIME HOTSPOT MODEL 75 Downloaded //4 to Redistribution subject to SIAM license or copyright; see From now on, we consider the case of two spikes having different amplitudes asymmetric spikes Using the notation φ ψ ω 3 ˆDA w φ Φ=, Ψ=, ω = = dy w, φ dy φ ψ we can rewrite 5 for x = t i as follows: where C = π ˆDA v 3/ s v 3/ l ψ B + CΨ = ω,, B = d 3 ˆDA, d = t t Using E = CB + C, we get the following system of nonlocal eigenvalue problems NLEPs: w 53 LΦ :=Φ yy Φ+3w Φ 3w 3 EΦ dy w3 dy Diagonalizing the matrix E, we know from [3, 9] that 53 has a nontrivial solution iff E has eigenvalue λ e = 3 Thus it remains to compute the matrix E and its eigenvalues We get E =B + CC = BC + I = ˆDA πd v 3/ s v 3/ l v 3/ s v 3/ l + I Then E has the eigenvector v m, =, T with eigenvalue e m, = ˆDA πd vs 3/ + v 3/ l + and the eigenvector v m, = v 3/ v 3 s +vl 3 l,vs 3/ T with eigenvalue e m, = 3 For nondegeneracy, the condition e m, 3 has to be satisfied, which is equivalent to ˆDA vs 3/ + v 3/ l πd Using the formulas for d,v s,v l, we compute ˆDA vs 3/ + v 3/ l πd = ˆDA vs + v l 3 3 v s v l + v s vl πl = ˆDA Ā A 3/ L 3 πl ˆDA 3 πl 3/ ˆDA = Ā A 3/ L 4π 3 ˆDA 4

16 76 HENRI BERESTYCKI, JUNCHENG WEI, AND MATTHIAS WINTER This implies the condition Downloaded //4 to Redistribution subject to SIAM license or copyright; see π ˆDA /4 Ā A 3/4 L 5 We have to exclude this point from our existence result Theorem 3 This is why we impose the condition in Theorem 3, which amounts to a nondegeneracy condition If this condition is violated, we expect small eigenvalues to occur, and whether there will be spikes in this case is an open question 6 Approximate solutions For simplicity, we set L = In this section and the following we consider the case of general K =,, since it does not cause any extra difficulty here, even in the caseofasymmetricspikeslet <t < <t j < t K < bek points and let v j > bek amplitudes satisfying the assumptions 4, 4, and 44 Let 6 t =t,,t K We first construct an approximate solution to 3 which concentrates near these prescribed K points Then we will rigorously construct an exact solution which is given by a small perturbation of this approximate solution Let <t < <t j < <t K < bek points such that t =t,,t K B 3/4t Set x tj 6 w j x =w and 63 r = min t +, t K, min i j t i t j Let χ : R [, ] be a smooth cut-off function such that χx =for x < and χx =for x > We now define the approximate solution as x tj 64 w j x =w j xχ It is easy to see that w j x satisfies 65 w j w j + w 3 j = est in L,, where est denotes an exponentially small term Let 66 Â = w,t x = j= 67 ˆv = T [w,t ], r v j w j x, where v j = T [w,t]t j, where T [A] is defined by 3 and t B 3/4t

17 EXISTENCE OF SPIKES FOR A CRIME HOTSPOT MODEL 77 Downloaded //4 to Redistribution subject to SIAM license or copyright; see Then by 33 we have 68 vi := T [Â]t i = lim T [w,t]t i +O log 69 Now let x = t i + y We find for  = w,t T [Â]t i + y T [Â]t i ti+y s = t i ˆD A +  y = ˆD A + w s + y s ˆvA + Â3 Ā + A dt ds ˆD A + w s d s +O log y = Pi y+ +O in L Ω, where w t 3 d td s v i w t 3 i d t + vi π t i + L vj L j= j= π vj Ht i t j ˆD A + w s d s i π π π + vi vj j= j= log P i y = y s ˆD A + w s w t 3 d td s v i vj t i + L L using 4 Note that Pi is an even function and the second term is an odd function in y We now derive the following estimate for all y : y T [w,t ]t i + y T [w,t ]t i w,t 3 C ˆD A + w s d sw3 y CˆD y w s d sw3 y CˆD y 3 e s d se 3y CˆD 3 e y For y< there is an obvious modification of this estimate Further, it can be extended to cover both of the cases when w 3,t is replaced by w,t or w,t, respectively, giving the same upper bound in either case

18 78 HENRI BERESTYCKI, JUNCHENG WEI, AND MATTHIAS WINTER Now if we define the norm Downloaded //4 to Redistribution subject to SIAM license or copyright; see 6 f = f L Ω + sup [maxmin e y t i, ] fy, L <y< L i then by the decay of w,t and the definition of the norm, we infer that 6 T [w,t ]t i + y T [w,t ]t i w 3,t = O 5/, 6 T [w,t ]t i + y T [w,t ]t i w,t = O 5/, 63 T [w,t ]t i + y T [w,t ]t i w,t = O 5/ Let us now define 64 S [Â] :=  xx  + T [Â]A + Â3, where T [Â] is defined in 3 Next we set  = w,t and compute S [w,t ] In fact, S [w,t ]= w,t xx w,t + T [w,t ]A + w,t 3 = w,t xx w,t + T [w,t ]t i w 3,t + T [w,t ] 3 A 3 +3 A w,t +3A w,t +T [w,t ]t i + y T [w,t ]t i w 3,t 65 =: E + E + E 3 We compute E = i= w v i w i + T [w,t]t i w i 3 = est i v i in L Ω sincev i = T [w,t]t i Further, we get E = O in L Ω sincet [w,t ] is bounded in L Ω andw,t is bounded in L Ω We also notice that actually E = Oe mini y t i Lastly, we derive E 3 = i= v i 3/ T [w,t]t i + y T [w,t ]t i w 3,i = O 3 in L Ω by 6 Combining these estimates, we conclude that 66 S[w,t ] = O Remark 8 The estimate 66 shows that our choice of approximate solution given in 66 and 67 is suitable This will enable us in the next two sections to rigorously construct a steady state which is very close to the approximate solution

19 EXISTENCE OF SPIKES FOR A CRIME HOTSPOT MODEL 79 7 Lyapunov Schmidt reduction In this section, we use Lyapunov Schmidt reduction to solve the problem Downloaded //4 to Redistribution subject to SIAM license or copyright; see 7 S [w,t + v] = i= β i d w i dx for real constants β i and a function v H, which is small in the corresponding norm to be defined later, where w i is given by 64 and w,t by 66 This is the first step in the Lyapunov Schmidt reduction method We shall follow the general procedure used in [5] To this end, we need to study the linearized operator L,t : H Ω L Ω given by 7 L,t := S [w,t]φ = Δφ φ+t [w,t ]3A +w,t φ+t [w,t ]φa +w,t 3, where for  = w,t and a given function φ L Ω we define T [Â]φ to be the unique solution of 73 ˆD A +  T [Â]φ x T [Â]φ x =T [Â]φ xl = x T [Â]φA + Â3 T [Â]3A +  φ = for L<x<L, The norm for the error function φ is defined as follows: 74 φ = φ H Ω + sup [maxmin e y t i, ] φy L <y< L i We define the approximate kernel and cokernel, respectively, as follows: K,t := span C,t := span { d w i dx } i =,,K H Ω, { } d w i dx i =,,K L Ω From 53 we recall the definition of the following system of NLEPs : 75 LΦ :=Φ yy Φ+3w Φ 3w 3 w EΦ dy w3 dy, where Φ= φ φ φ K H R K

20 7 HENRI BERESTYCKI, JUNCHENG WEI, AND MATTHIAS WINTER By Lemma 33 of [5] we know that Downloaded //4 to Redistribution subject to SIAM license or copyright; see L :X X H R K X X L R K is invertible and its inverse is bounded We will show that this system is the limit of the operator L,t defined in 7 as We also introduce the projection π,t : L Ω C,t and study the operator L,t := π,t L,t By letting, we will show that L,t : K,t C,t is invertible and its inverse is uniformly bounded provided is small enough This statement is contained in the following proposition Proposition 9 There exist positive constants, δ, λ such that for all,, t Ω K with min +t, t K, min i j t i t j > δ, 76 L,t φ λ φ Furthermore, the map L,t = π,t L,t : K,t C,t is surjective Proof This proof uses the method of Lyapunov Schmidt reduction following, for example, the approach in [, 5, 7] Suppose that 76 is false Then there exist sequences { k }, {t k }, {φ k } with k, t k Ω K, min +t k, tk K, min i j t k i tk j > δ, φ k = φ k K k,t, k k =,,, such that L k,t kφk as k, φ k =, k =,, We define φ,i, i =,,,K,andφ,K+ as follows: x ti 79 φ,i x =φ xχ, x Ω, φ,k+ x =φ x r φ,i x, x Ω At first after rescaling the functions φ,i are only defined on Ω However, by a standard result they can be extended to R such that their norm in H R is still bounded by a constant independent of and t for small enough In the following we will deal with this extension For simplicity of notation we keep the same notation for the extension Since for i =,,,K each sequence {φ k i } := {φ k,i} k =,, is bounded in Hloc R it converges weakly to a limit in H loc R, and therefore also strongly in L loc R andl loc R Denoting these limits by φ i,then φ = solves the system Lφ = By Lemma 33 of [5], it follows that φ KerL = X X Sinceφ k K k,t k, taking k,wegetφ KerL Therefore, we have φ = i= φ φ φ K

21 EXISTENCE OF SPIKES FOR A CRIME HOTSPOT MODEL 7 Downloaded //4 to Redistribution subject to SIAM license or copyright; see By elliptic estimates we get φ k,i H R ask for i =,,,K Further, φ,k+ φ K+ in H R, where Φ K+ satisfies φ K+ yy φ K+ = inr Therefore, we conclude that φ K+ =and φ k K+ H R ask Once we have φ i H R, the maximum principle implies that φ i since the operator L,t essentially behaves like φ i φ i for x t i This contradicts the assumption that φ k = To complete the proof of Proposition 9, we just need to show that the conjugate operator to L,t denoted by L,t is injective from K,t to C,t NotethatL,tφ = π,t L,tφ with L,t φ := Δφ φ + T [w,t ]3A + w,t φ +T [w,t ]φa + w,t 3 The proof for L,t follows along the same lines as the proof for L,t and is therefore omitted Here also the nondegeneracy condition is required For further technical details we refer the reader to [5] Now we are in a position to solve the equation 7 π,t S w,t + φ = Since L,t K,t is invertible call the inverse L,t, we can rewrite this equation as 7 φ =,t π,t S w,t L,t π,t N,tφ M,t φ, where 7 N,t φ =S w,t + φ S w,t S w,t φ and the operator M,t has been defined by 7 for φ H Ω The strategy of the proof is to show that the operator M,t is a contraction on B,δ {φ H Ω : φ <δ} if is small enough and δ is suitably chosen By 66 and Proposition 9 we have that M,t φ λ π,t N,tφ + π,t S w,t λ C cδδ +, where λ> is independent of δ>, >, and cδ asδ Similarly, we show that M,t φ M,t φ λ C cδδ φ φ, where cδ asδ Choosing δ = C 3 for λ C <C 3 and taking small enough, then M,t maps B,δ into B,δ, and it is a contraction mapping in B,δ The existence of a fixed point φ,t now follows from the standard contraction mapping principle and φ,t is a solution of 7 We have thus proved the following Lemma There exist > δ>such that for every pair of, t with << and t Ω K, +t > δ, t K > δ, t i t j > δ there is a unique φ,t K,t satisfying S w,t + φ,t C,t Furthermore, the following estimate holds: 73 φ,t C 3 In the next section we determine the positions of the spikes so that the resulting steady state is an exact solution of the original problem

22 7 HENRI BERESTYCKI, JUNCHENG WEI, AND MATTHIAS WINTER Downloaded //4 to Redistribution subject to SIAM license or copyright; see 8 The reduced problem In this section we solve the reduced problem and complete the proof of the existence result for asymmetric spikes in Theorem 3 By Lemma, for every t B 3/4t, there exists a unique φ,t K,t solution of 8 S[w,t + φ,t ]=v,t C,t The idea here is to find t =t,,t K neart such that also 8 S[w,t + φ,t ] C,t and therefore S[w,t + φ,t ]= To this end, we let W,i t := v i S[w,t + φ,t ] d w i dx dx, W t :=W, t,,w,k t : B 3/4t R K Then W t is a map which is continuous in t, and our problem is reduced to finding a zero of the vector field W t Let us now calculate W t as follows: W,i t = v i = v i + v i + v i S[w,t + φ,t ] d w i dx S[w,t ] d w i dx L =: I + I + I 3, S [w,t]φ,t d w i dx N φ,t d w i dx where I,I,andI 3 are defined in an obvious way in the last equality We will now compute these three integral terms as The result will be that I is the leading term and I and I 3 are O log For I,wehave I = v i L E + E + E 3 d w i dx dx = v i L d w i E 3 dx dx + O log, where E,E,E 3 are defined in 65 For E this estimate is obvious For E,we use the decomposition T [w,t ] 3 A 3 +3 A w,t +3A w,t = T [w,t ] 3 A 3 + T [w,t ]t i 3 A w,t +3A w,t +T [w,t ]t i + y T [w,t ]t i 3 A w,t +3A w,t

23 EXISTENCE OF SPIKES FOR A CRIME HOTSPOT MODEL 73 Downloaded //4 to Redistribution subject to SIAM license or copyright; see Then we can estimate the first part directly, the second part using the fact that it is an even function in y, and the third part using the estimates 6 and 63 From 6, we derive v L i = v i L/ + v i / E 3 d w i dx dx P i yw 3 y w y vi j= dy i π π π + vi vj j= vj t i + L L L/ y d sa + w y / ˆD Â3 χ i dy A + Â s vi + O log = est ˆD i π π π t i + L + + O log, vi vj vj L j= where χ i x =χ x ti r Here we have used the fact that P i y is an even function and have computed the following integral: = L/ y / L/ y / = ˆD = ˆD L/ / R = ˆDv i R = ˆDv i + O j= y d sa + ˆD Â3 w χ i t i + y dy A + Ât vi i + s d s d ˆD A + Ât 4 dy A + Â4 χ i t i + y dy + O i + s A + 4 Ât i + y dy + O 3 log dy + O 3 log wy 4v i cosh y dy + O 3 log In summary, we have 3 log 83 I = ˆD i π π π t i + L + + O log vi vj vj L j= j= 3 log

24 74 HENRI BERESTYCKI, JUNCHENG WEI, AND MATTHIAS WINTER For I,wecalculate Downloaded //4 to Redistribution subject to SIAM license or copyright; see I = v i = v i L = v L i + v i + v i + v i + v i + v i L S [w,t ]φ,t d w i dx [ Δφ,t φ,t + T [w,t ]3A + w,t φ,t +T [w,t ]φ,t A + w,t 3] d w i dx [ Δ d w i L L L L L dx d w i dx +3 w i T [w,t ]x T [w,t ]t i v i T [w,t ]t i 3 A +A T [w,t ]t i d w i v i dx T [w,t ]x T [w,t ]t i 3 3 w i d w i φ,t dx dx w i vi ] φ,t dx φ,t d w i dx dx A +A w i vi T [w,t ]φ,t 3 A 3 d w i dx dx T [w,t ]φ,t t i 3 A w i w i +3A + w i vi v i + v i [T [w,t ]φ,t x T [w,t ]φ,t t i ] 3 A 3 w i w i +3A + w i d w i vi v i v 3/ dx dx i = I + I + I 3 + I 4 + I 5 + I 6 + I 7 v 3/ i φ,t d w i dx d w i dx dx With obvious notation, we now show that each one of the seven terms is O log as For I, it follows from T [w,t] =v i, while for I, we use 6 and the fact that 84 φ,t C L Ω w,t For I 3, we use T [w,t ] L Ω = O and the fact that φ,t is an even function For I 4, we use 6, 63, and 84 For I5, the estimate is derived from T [w,t ]φ,t L Ω = O For I 6,weuseT [w,t ]φ,t t i =O andthefact that w i is even Lastly, for I 7, we use estimates similar to 6, 6, 63 with T [w,t ]φ,t instead of T [w,t ] and the inequality T [w,t ]φ,t T [w,t ] C L Ω By arguments similar to those for I, we derive 85 I 3 = O log in L Ω

25 EXISTENCE OF SPIKES FOR A CRIME HOTSPOT MODEL 75 Downloaded //4 to Redistribution subject to SIAM license or copyright; see Combining the estimates for I, I,andI 3,wehave W,i t = ˆD i π π π + vi vj j= j= = ˆD F i t+o log 86, vj t i + L L + O where F i t was defined in 43 By assumption 44, we have F t = Next we show that det t F t log in the case of two spikes with amplitudes v = v s <v l = v We compute where D t v = t F t =D t F +D v F D t v, D t F = π I, vj L π D v F = 4 π ˆDA 3/ v v + This implies, using 4 and 46, that π t F t = 4 ˆDA v v 3/ + j= v 3/ v 3/ πl ˆDA v 3/ πl ˆDA v 3/, v / v / v v3/ v 3/ v v 5/ v / v v +v +v 3/ v / v 3/ v / Next, we compute det t F t = v v 5/ v / v v +v +v 3/ v / = π 4 6 ˆDA v v α 3 α α + α 3 α α +α 3 α α + α 3 π 4 6 ˆDA v v α 3 α α + α 3 α α +,

26 76 HENRI BERESTYCKI, JUNCHENG WEI, AND MATTHIAS WINTER Downloaded //4 to Redistribution subject to SIAM license or copyright; see where α = v v Therefore, we have det t F t, except for two specific positive values of α: α = the bifurcation point of asymmetric from symmetric spikes which is not included in Theorem 3 and α = + 5 corresponding to the eigenvalue e m, = 3 in section 5 which has been excluded from Theorem 3 Thus, under the conditions of Theorem 3, we get W t = ˆD t F t t t +O t t + log Since W t is continuous in t, standard degree theory [6] implies that for small enough and δ suitable chosen there exist t B δ t such that W t = and t t For further technical details of the argument, see [7] Thus we have proved the following proposition Proposition For small enough, there exist points t with t t such that W t = Finally, we complete the proof of Theorems and 3 ProofofTheorem3 By Proposition, there exist t t such that W t = In other words, S[w,t + φ,t ]= Let = w,t + φ,t, ˆv = T [w,t + φ,t ] By the maximum principle, we conclude that  >, ˆv > Moreover Â, ˆv satisfies all the properties of Theorem 3 Proof of Theorem To prove Theorem, we first construct a single spike in the interval L K, L K as above Then we continue the single spike periodically to a function in the interval, L and get a symmetric multiple spike in the interval, L 9 Proof of Theorem 4 The proof of Theorem 4 goes exactly as that of Theorem 3 First, let us derive the location of the single spikes formally: In the first equation in 3 the term 3 A is very small and can be omitted in the computations Thus we may assume that x 9  ξ / t w, vt =ξ Substituting the above expressions into the second equation of 3 and noting that ˆv A + Â3 ξ / w 3 δ t,weseethatˆv satisfies in leading order: 9 ˆDA ˆv x x ξ / w 3 dy δ t + γx = Solving the above equation, we then obtain 93 ˆv x t = ˆDA t t R γxdx, ˆv x t + = L γxdx ˆDA t Substituting 9 into the first equation of 3 and rescaling x = t + y, we deduce that the error becomes 94 ˆv x t y +ˆv x t +y + w 3 y+a t 3w + O, where y = miny, and y + =maxy,, from which we conclude that a necessary condition for the existence of a spike at t is that [ 95 ˆvx t y +ˆv x t +y + w 3 y+a t 3w + O ] w y dy =, R t

27 EXISTENCE OF SPIKES FOR A CRIME HOTSPOT MODEL 77 whence Downloaded //4 to Redistribution subject to SIAM license or copyright; see 96 ˆv x t +ˆv x t + =, which is equivalent to 7 It turns out that 7 is also sufficient, since the derivative of t γxdx L t γxdx with respect to t is γt, which is strictly positive The rest of the proof matches exactly the proof of Theorem 3 We omit the details Discussion In this article we have provided a rigorous mathematical analysis of the formation of spikes in the model of Short, Bertozzi, and Brantingham [9] Thus, we have shown that this model naturally leads to the formation of criminality hotspots The existence of such hotspots is one of the main stylized facts about criminality It is observed for an array of criminal activity types Hotspots are extensively reported and discussed in the criminology literature See, for example, the articles [9] and [4] as well as the references therein Now, the fact that a mathematical model yields such hotspots can be viewed as passing one benchmark of validity The findings in our paper provide such a test for the Short, Bertozzi, and Brantingham model [9] Furthermore, the rigorous analysis carried here sheds light on the mechanisms for the formation of hotspots in this model and the way it quantitatively depends on the parameters This type of analysis can then be applied to study issues such as the reduction of hotspots by crime prevention strategies or optimal use of resources to this effect One of the goals is to understand when policing strategies actually reduce criminal activity and when they merely displace hotspots to new areas In this paper we have proved three main new results First, we showed that we can reduce the quasilinear chemotaxis problems to a Schnakenberg-type reactiondiffusion system and derived the existence of symmetric k spikes Next, we established the existence of asymmetric spikes in the isotropic case Last, we have studied the pinning effect by inhomogeneous media A x andγx The stability of these spikes is an interesting open issue In [3] spikes in two space dimensions are considered by formal matched asymptotics Our approach of rigorous justification can be extended to that case in a radially symmetric setting, ie, if the domain is a disk and we construct a single spike located at the center We remark that in [3] it is assumed that in the outer region away from the spikes the system in leading order is semilinear, which allows an extension of the results for the Schnakenberg model to this case In [3], for the inner region, a numerical computation by solving a core problem yields the profile of the spike An alternative approach to quasilinear problems in one space dimension would be to write them as first-order semilinear ODE systems and then apply standard methods, eg, dynamical systems methods for the problem on the real line We refer the reader to a recent paper [], where the dynamical systems approach is used to construct traveling wave solutions of a quasilinear reaction-diffusion mechanical system We remark that there are very few results concerning the analysis of spikes in quasilinear reaction-diffusion systems As far as we know, there are two such types of systems The first one is the chemotaxis system of Keller Segel type We refer the reader to [] for the background of chemotaxis models and [5] for the analysis of spikes to these systems The other one is the Shigesada Kawasaki Teramoto model of species segregation [8] For the analysis of spikes in a cross-diffusion system, see [4, 6, 3]

28 78 HENRI BERESTYCKI, JUNCHENG WEI, AND MATTHIAS WINTER Downloaded //4 to Redistribution subject to SIAM license or copyright; see A family of related models for the diffusion of criminality has been proposed in [] We analyze the formation of hotspots in this class of models in our forthcoming work [3] The equations in [] also envision the possibility of nonlocal diffusion Indeed, social influence can be exercised at long range, and it is natural to consider descriptions that take long range diffusion into account Such a nonlocal system arising in [] reads { s t x, t =Lsx, t sx, t+s b + αxux, t, u t x, t =Λs ux, t The case when L = Δ is a local diffusion operator provides the framework of the study in [] Here, L can also be a nonlocal operator such as the fractional Laplace operator or a general nonlocal interaction term: Lsx, t = Jx, ysy, t sx, tdy Observe that the steady states reduce to a single nonlocal equation: s = s b x s + αxλs We note that the interaction between nonlocal diffusion and the mechanism for the formation of spikes is completely open In particular, the description of the formation of spikes in and are open problems We expect that the decay of the kernel may come into play for the formation of spikes Acknowledgments Part of this work was done while Henri Berestycki was visiting the Department of Mathematics at University of Chicago Matthias Winter thanks the Department of Mathematics of The Chinese University of Hong Kong for its kind hospitality Lastly, the authors are thankful to the referees for careful reading of the manuscript and many constructive suggestions REFERENCES [] H Berestycki and J Nadal, Self-organised critical hotspots of criminal activity, European J Appl Math,, pp [] H Berestycki, N Rodríguez, and L Ryzhik, Traveling wave solutions in a reactiondiffusion model for criminal activity, Multiscale Model Simul, 3, pp 97 6 [3] H Berestycki, M Winter, and J Wei, Analysis of the Formation of Crime Hotspots for a Model Proposed by Berestycki and Nadal, manuscript [4] P Brantingham and P Brantingham, Theoretical model of crime hot spot generation, Studies on Crime and Crime Prevention, 8 999, pp 7 6 [5] R S Cantrell, C Cosner, and R Manásevich, Global bifurcation of solutions for crime modeling equations, SIAM J Math Anal, 44, pp [6] E N Dancer, Degree theory on convex sets and applications to bifurcation, incalculusof Variations and Partial Differential Equations: Topics on Geometrical Evolutions Problems and Degree Theory, L Ambrosio and E N Dancer, eds, Springer, New York,, pp 85 5 [7] A Doelman, R A Gardner, and T J Kaper, Large stable pulse solutions in reactiondiffusion equations, Indiana Univ Math J, 49, pp 3 45 [8] A Doelman, T J Kaper, and H van der Ploeg, Spatially periodic and aperiodic multipulse patterns in the one-dimensional Gierer-Meinhardt equation, Methods Appl Anal, 8, pp [9] J Eck, Crime hot spots: What they are, why we have them, and how to map them, in Mapping Crime: Understanding Hotspots, J Eck, S Chainey, J Cameron, and R Wilson, eds, National Institute of Justice, Washington, DC, 5, Chap [] T Hillen and K J Painter, A user s guide to PDE models for chemotaxis, J Math Biol, 58 9, pp 83 7

29 EXISTENCE OF SPIKES FOR A CRIME HOTSPOT MODEL 79 Downloaded //4 to Redistribution subject to SIAM license or copyright; see [] M Holzer, A Doelman, and T J Kaper, Existence and stability of traveling pulses in a reaction-diffusion-mechanics system, J Nonlinear Sci, 3 3, pp 9 77 [] D Iron, J Wei, and M Winter, Stability analysis of Turing patterns generated by the Schnakenberg model, J Math Biol, 49 4, pp [3] T Kolokolnikov, M J Ward, and J Wei, The stability of steady-state hot-spot patterns for a reaction-diffusion model of urban crime, Discrete Contin Dyn Syst Ser B, to appear [4] T Kolokolnikov and J Wei, Stability of spiky solutions in a competition model with crossdiffusion, SIAM J Appl Math, 7, pp [5] T Kolokolnikov and J Wei, Basic mechanisms driving complex spike dynamics in a chemotaxis model with logistic growth, SIAM J Appl Math, submitted [6] Y Lou and W-M Ni, Diffusion vs cross-diffusion: An elliptic approach, J Differential Equations, , pp 57 9 [7] N Rodriguez and A Bertozzi, Local existence and uniqueness of solutions to a PDE model for criminal behavior, Math Models Methods Appl Sci,, suppl, pp [8] N Shigesada, K Kawasaki, and E Teramoto, Spatial segregation of interacting species, J Theoret Biol, , pp [9] M B Short, A L Bertozzi, and P J Brantingham, Nonlinear patterns in urban crime: Hotspots, bifurcations, and suppression, SIAM J Appl Dyn Syst, 9, pp [] M B Short, A L Bertozzi, P J Brantingham, and G E Tita, Dissipation and displacement of hotspots in reaction-diffusion model of crime, Proc Natl Acad Sci USA, 7, pp [] M B Short, M R D Orsogna, V B Pasour, G E Tita, P J Brantingham, A L Bertozzi, and L B Chayes, A statistical model of crime behavior, Math Methods Appl Sci, 8 8, pp [] M J Ward and J Wei, Asymmetric spike patterns for the Schnakenberg model, Stud Appl Math, 9, pp 9 64 [3] J Wei, On single interior spike solutions of Gierer-Meinhardt system: Uniqueness, spectrum estimates, European J Appl Math, 999, pp [4] J Wei, Existence and stability of spikes for the Gierer-Meinhardt system, in Handbook of Differential Equations: Stationary Partial Differential Equations, Handb Differ Equ 5, M Chipot, ed, Elsevier, Amsterdam, 8, pp [5] J Wei and M Winter, Existence, classification and stability analysis of multiple-peaked solutions for the Gierer-Meinhardt system in R, Methods Appl Anal, 4 7, pp 9 63 [6] J Wei and M Winter, On the Gierer-Meinhardt system with precursors, Discrete Contin Dyn Syst, 5 9, pp [7] J Wei and M Winter, Stationary multiple spots for reaction-diffusion systems, J Math Biol, 57 8, pp [8] J Wei and M Winter, Mathematical Aspects of Pattern Formation in Biological Systems, Appl Math Sci 89, Springer, London, 4 [9] J Wei and L Zhang, On a nonlocal eigenvalue problem, Ann Scuola Norm Sup Pisa Cl Sci, 3, pp 4 6 [3] Y Wu and Q Xu, The existence and structure of large spiky steady states for S-K-T competition systems with cross-diffusion, Discrete Contin Dyn Syst, 9, pp