Cops-on-the-Dots: The Linear Stability of Crime Hotspots for a 1-D Reaction-Diffusion Model of Urban Crime

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1 Cops-on-the-Dots: The Linear Stability of Crime Hotspots for a 1-D Reaction-Diffusion Model of Urban Crime Andreas Buttenschoen, Theodore Kolokolnikov, Michael J. Ward, Juncheng Wei October 7, 18 Abstract In a singularly perturbed limit, e analyze the existence and linear stability of steady-state hotspot solutions for an extension of the 1-D three-component reaction-diffusion (RD) system formulated and studied numerically in Jones et. al. [Math. Models. Meth. Appl. Sci.,, Suppl., (1)], hich models urban crime ith police intervention. In our extended RD model, the field variables are the attractiveness field for burglary, the criminal density, and the police density, and it includes a scalar parameter that determines the strength of the police drift toards maxima of the attractiveness field. For a special choice of this parameter, e recover the cops-on-the-dots policing strategy of Jones et. al., here the police mimic the drift of the criminals toards maxima of the attractiveness field. For our extended model, the method of matched asymptotic expansions is used to construct 1-D steady-state hotspot patterns as ell as to derive nonlocal eigenvalue problems (NLEPs), each having three distinct nonlocal terms, that characterize the linear stability of these hotspot steady-states to O(1) time-scale instabilities. For a cops-on-the-dots policing strategy, here some key identities can be used to recast these NLEPs into equivalent NLEPs ith only one nonlocal term, e prove that a multi-hotspot steady-state is linearly stable to synchronous perturbations of the hotspot amplitudes. Moreover, for asynchronous perturbations of the hotspot amplitudes, a hybrid analytical-numerical method is used to construct linear stability phase diagrams in the police versus criminal diffusivity parameter space. In one particular region of these phase diagrams, the hotspot steady-states are shon to be unstable to asynchronous oscillatory instabilities in the hotspot amplitudes that arise from a Hopf bifurcation. Within the context of our model, this provides a parameter range here the effect of a cops-on-the-dots policing strategy is to only displace crime temporally beteen neighboring spatial regions. Our hybrid approach to study the equivalent NLEP combines rigorous spectral results ith a numerical parameterization of any Hopf bifurcation threshold. For the cops-on-the-dots policing strategy, our linear stability predictions for steady-state hotspot patterns are confirmed from full numerical PDE simulations of the three-component RD system. Key Words: Urban crime, hotspot patterns, nonlocal eigenvalue problem (NLEP), Hopf bifurcation, asynchronous oscillatory instability, cops-on-the-dots. 1 Introduction Motivated by the increased availability of residential burglary data, the development of mathematical modeling approaches to quantify and predict spatial patterns of urban crime as initiated in [18 ]. One primary feature incorporated into these models, hich is based on observations from the available data (cf. []), is that spatial patterns of residential burglary are typically concentrated in small regions knon as hotspots; a feature believed to be attributable to a repeat or near-repeat victimization effect (cf. [9], []). There have been to primary frameorks that have been used to model the effect of police intervention on crime hotspot patterns. One approach, ideal for incorporating detailed real-orld policing strategies, is large scale simulations of agent-based models (cf. [1], [5]). Hoever, ith this approach, it is difficult to isolate the effect of Dept. of Mathematics, UBC, Vancouver, Canada. abuttens@math.ubc.ca Dept. of Mathematics and Statistics, Dalhousie, Halifax, Canada. tkolokol@gmail.com Dept. of Mathematics, UBC, Vancouver, Canada. (corresponding author ard@math.ubc.ca) Dept. of Mathematics, UBC, Vancouver, Canada. jcei@math.ubc.ca 1

2 changes in the model parameters. A second, more phemenological approach, is to formulate PDE-based reaction-diffusion (RD) systems that model the police density as an extra field variable (cf. [1], [15], [16]). More elaborate PDE models, such as in [1], formulate an optimal control strategy to minimize the overall crime rate by alloing the police to adapt to dynamically evolving crime patterns. In our PDE-based approach, motivated by [1] and [16], the police intervention is modeled by a drift-diffusion PDE, in hich a parameter models the strength of the drift toards the maxima of the attractiveness field for burglary. For this three-component RD system consisting of an attractiveness field coupled to the criminal and police densities, e ill study the existence and linear stability properties of steady-state hotspot patterns on a 1-D spatial domain < x < S in a singularly perturbed limit. The specific three-component RD model of urban crime that e ill analyze is formulated as A t = ǫ A xx A+ρA+α, ρ t = D(ρ x ρa x /A) x ρa+γ α ρu, τu t = D(U x qua x /A) x, (1.1a) (1.1b) (1.1c) here A x = ρ x = U x = at x =,S. Here A is the attractiveness field for burglary, hile ρ and U are the densities of criminals and police, respectively. In this model, α is the baseline attractiveness, γ α > is the rate at hich ne criminals are introduced, D is the criminal diffusivity, D p D/τ is the police diffusivity, and ǫ 1 characterizes the repeat or near-repeat victimization effect (cf. [18], [9], []). For (1.1c), the total policing level U is prescribed, and it satisfies U S U(x,t)dx, (1.) so that it is conserved in time. The model parameters α, γ α, D, D p, and U are all assumed to be positive constants. In (1.1), the parameter q > measures the degree of focus in the police patrol toards maxima of the attractiveness field. The choice q =, hich recovers the PDE system derived and studied numerically in [1], is the cops-on-the-dots strategy (cf. [1], [16]) here the police mimic the drift of the criminals toards maxima of A. In (1.1b), the police density at a given spatial location decreases the local criminal population at a rate proportional to the local criminal density (the ρu term in (1.1b)). The resulting predator-prey type police interaction model (1.1) is to be contrasted ith the simple police interaction model formulated in [16], and analyzed in [], here the ρu term in (1.1b) is replaced by U. Intheabsenceofpolice, i.e. U =, (1.1)reducestotheto-componentPDEsystemforAandρfirstderivedandstudied in [18] and []. Pattern formation aspects for this basic crime model have been ell-studied from various viepoints, including, eakly nonlinear theory (cf. [19]), bifurcation theory near Turing points (cf. [6], [8]) and the computation of global snaking-type bifurcation diagrams (cf. [1]), rigorous existence theory (cf. [17]), and asymptotic methods for constructing steady-state hotspot patterns hose linear stability properties can be analyzed via NLEP theory (cf. [11], [1], [1]). Our goal here is to extend the analysis given in [] for the existence and linear stability of hotspot steady-states for the simple police interaction model to the predator-prey type interaction model (1.1). We ill sho that the seemingly minor and innocuous replacement of U from the model in [] ith ρu in (1.1b) leads to a significantly more challenging linear stability problem for hotspot equilibria. This is discussed in detail belo. As in [] and [11], e ill analyze (1.1) in the limit ǫ for the range D = O(ǫ ). Since A = O(ǫ 1 ) in the core of the hotspot, it convenient as in [] to introduce the ne variables v and u by ρ = ǫ va, U = ua q, D = ǫ D. (1.)

3 In terms of A, v, and u, on the domain < x < S, and ith no flux boundary conditions at x =,S, (1.1) transforms to A t = ǫ A xx A+ǫ va +α, (1.a) ǫ ( A v ) t = D( A v x )x ǫ va +γ α ǫ uva +q, (1.b) τǫ (A q u) t = D(A q u x ) x, here the police diffusivity D p becomes D p = ǫ D/τ. In e use a formal singular perturbation analysis in the limit ǫ to construct hotspot steady-state solutions to (1.) that have a common amplitude. Our steady-state analysis is restricted to the range q > 1, for hich the police density is asymptotically small in the background region aay from the hotspots. In Proposition.1 and Corollary. belo e establish that steady-state hotspot solutions exist only hen U < U,max S(γ α)(q +1)/(q). In e use a singular perturbation analysis combined ith Floquet theory, similar to but more intricate than that used in [] and [11], to derive to distinct NLEP spectral problems characterizing the linear stability of hotspot steady-states of (1.). One such NLEP, given belo in Proposition., characterizes the linear stability properties of a multi-hotspot steady state solution, having K hotspots, to synchronous perturbations in the hotspot amplitudes. The linear stability properties of a one-hotspot steady-state is also determined by the spectrum of this NLEP. In addition, the second NLEP, given belo in Proposition., characterizes the linear stability properties of a multi-hotspot steady-state, ith K hotspots, to K 1 > different spatial modes of asynchronous perturbations of the hotspot amplitudes. A complicating feature in the analysis of these spectral problems is that each of the to NLEPs has three distinct nonlocal terms consisting of a linear combination of Φ, q 1 Φ, and q+1 Φ. Here (y) = sechy is the homoclinic profile of a hotspot, and Φ(y) is the NLEP eigenfunction. As a result of this complexity, the determination of unstable spectra for these NLEPs is seemingly beyond the general NLEP stability theory ith a single nonlocal term, as surveyed in [9]. For the simple police interaction model, studied in [], the corresponding NLEPs had only to nonlocal terms. In e use a hybrid analytical-numerical strategy to determine the spectrum of the NLEP characterizing the linear stability to synchronous perturbations. For arbitrary q > 1, the to different approaches developed in.1 and. provide clear numerical evidence that this NLEP has no unstable eigenvalues. This strongly indicates that, for any q > 1, a onehotspot steady-state is alays linearly stable and that a multi-hotspot steady-state is alays linearly stable to synchronous perturbations in the hotspot amplitudes. For the special case q = of cops-on-the-dots, in..1 this linear stability conjecture is proved rigorously. This proof of linear stability for q = relies on some key identities that allo the NLEP ith three nonlocal terms to be converted into an equivalent NLEP ith a single nonlocal term. For general q > 1, in 5 e determine the threshold value of D corresponding to a zero-eigenvalue crossing of the NLEP, as defined in Proposition., that characterizes the linear stability of a multi-hotspot steady-state to the asynchronous modes. For a K-hotspot steady-state ith K, this critical value of D, called the competition stability threshold, is D c (1.c) S [ (1 q)ω 8K π α +qs(γ α)ω ], here ω S(γ α) qu /(q +1), (1.5) [1+cos(π/K)] onu < U,max S(γ α)(q +1)/(q). Inthelimitingcaseofaninfinitepolicediffusivity, aindingnumberanalysisisused in 5.1 to prove, for an arbitrary q > 1, that a multi-hotspot steady-state is linearly stable to asynchronous perturbations in the hotspot amplitudes if and only if D < D c (see Proposition 5. belo). For the special case q = of cops-on-the-dots, in 6 e sho ho to transform the NLEP for the asynchronous modes into an equivalent NLEP ith only one nonlocal term, hich is then more readily analyzed. With this reduction of the NLEP into a more standard form, hich only applies hen q =, in Proposition 6. e prove that a K-hotspot

4 steady-state, ith K, is alays unstable to the asynchronous modes hen D > D c for any police diffusivity D p >. Moreover, from a numerical parameterization of branches of purely complex eigenvalues for this equivalent NLEP, e sho that each of the K 1 asynchronous modes can undergo a Hopf bifurcation at some critical values of the police diffusivity D p defined on some intervals of D. Overall, this hybrid approach provides phase diagrams in the ǫd p versus D parameter plane characterizing the linear stability of the hotspot steady-states to asynchronous perturbations in the hotspot amplitudes. Numerical evidence from PDE simulations suggests that hotspot amplitude oscillations arising from the Hopf bifurcation can be either subcritical or supercritical, depending on the parameter set. Linear stability phase diagrams for various U are shon belo in Fig. 9 and Fig. 1 for K = and K =, respectively. One key qualitative feature derived from these phase diagrams is that there is a region in the ǫd p versus D parameter space here the effect of police intervention is to only displace crime temporally beteen neighboring spatial regions; a phenomenon qualitatively consistent ith the field observations reported in [] for a cops-on-the-dots policing strategy. As in [], e emphasize that the interval in D here asynchronous hotspot amplitude oscillations occur disappears hen U =. Therefore, it is the third component of the RD system (1.) that is needed to support these temporal oscillations. In contrast, for most to-component RD systems ith localized spike-type solutions, such as the the Gray-Scott and Gierer- Meinhardt models (cf. [7], [1], [], [1]), the dominant Hopf stability threshold for spike amplitude oscillations, based on an NLEP linear stability analysis, is determined by the spatial mode that synchronizes the oscillations. For q =, in 7 e validate the predictions of our linear stability analysis ith full numerical PDE simulations of (1.). Finally, in 8 e compare our linear stability results for (1.) for a cops-on-the-dots strategy ith those in [] for the simple police interaction model. We also briefly discuss some specific open problems and ne directions arranting study. Asymptotic Construction of a Multiple Hotspot Steady-State In the limit ǫ, e use the method of matched asymptotic expansions to construct a steady-state solution to (1.) on x S ith K 1 interior hotspots of a common amplitude. We follo the approach in [] in hich e first construct a one-hotspot solution to (1.) centered at x = on the reference domain x l. From translation invariance, this construction yields a K interior hotspot steady-state solution on the original domain of length S = (l)k. On x l, (1.) yields that l l U dx = U /K, here U is the constant total police deployment. On the reference domain x l, e center a steady-state hotspot at x =, and e impose A x = v x = u x = at x = ±l. For this canonical hotspot problem, the steady-state problem for (1.) is to find A(x), v(x), and the constant u, satisfying ǫ A xx A+ǫ va +α =, x l; A x =, x = ±l, (.1a) D ( A v x )x ǫ va +γ α ǫ uva +q =, x l; v x =, x = ±l, (.1b) here the steady-state police density U(x) is related to u by U = ua q, here u = U K l l Aq dx. (.) For ǫ, e have A α+o(ǫ ) in the outer region, hile in the inner region near x =, e set y = ǫ 1 x and expand A ǫ 1 A and v v in (.1). To leading-order, in the inner region e obtain from (.1) that A (y) v, v v. (.)

5 Here v is a constant to be determined and (y) = sechy is the homoclinic solution of + =, < y < ; () >, () =, as y ±. (.) Integrals of various poers of (y), as needed belo, can be calculated in terms of the Gamma function Γ(z) by (see [] and the appendix in []) I q q dy = q/ 1[Γ(q/)] Γ(q). (.5) We ill consider the range q > 1 here the dominant contribution to the integral l l Aq dx arises from the inner region: l A q dx ǫ 1 q v q/ q dy = O(ǫ 1 q ) 1. l Since q > 1, (.) shos that u depends to leading-order only on the inner region contribution from A q. For ǫ 1, e get u ǫ q 1 ũ e, here ũ e U v q/. (.6) KI q To determine v, e integrate (.1b) over l < x < l, hile imposing v x (±l) =. This yields that l ǫ va dx = l(γ α) ǫ u l l l va +q dx. (.7) Since A α = O(1) and A = O(ǫ 1 ) in the outer and inner regions, respectively, it follos that, hen q > 1, the dominant contribution to the integral arises from the inner region here v v. In this ay, and by using (.) in (.7), e get l dy l(γ α) ǫ U v l A+q dx v K l l Aq dx. (.8) By using A ǫ 1 (y)/ v, together ith (.5), e calculate the integral ratio in (.8) for ǫ as l l A+q dx l l Aq dx ǫ q+ dy v q dy = ǫ (q+)/ 1 ( ) Γ(1+q/) Γ(q) v q/ 1 Γ(q/) Γ(q +) = ǫ q v q +1, (.9) by using Γ(x+1) = xγ(x). Then, by substituting (.9) into (.8), and using dy = π, e solve for v to get [ v = π l(γ α) U ] q, (.1) K q +1 provided that the total level U of police deployment is belo a threshold given by U < U,max lk(γ α) (q +1) q (q +1) = S(γ α). (.11) q Here S = lk is the original domain length. We ill assume that (.11) holds, so that a K-hotspot steady-state exists. The amplitude of the hotspot, defined by A max A() 1, is given by A max A() ǫ 1 A () = ǫ 1() = ǫ 1 ω v πk, here ω S(γ α) U q q +1. (.1) This hotspot amplitude decreases ith increasing either K, U, or q. To complete the asymptotic construction of the hotspot, in the outer region e expand v v e (x) +... and use A α+o(ǫ ). From (.1b), e obtain to leading order that v e (x) satisfies (γ α) Dv exx = α, l < x < l; v e () = v, v ex (±l) =, (.1) hich is readily solved analytically. We summarize our leading-order results for a steady-state K-hotspot pattern as follos: 5

6 Proposition.1 Let ǫ, q > 1, and < U < U,max, here U,max is given in (.11). Then, (1.) admits a steadystate solution on (,S) ith K interior hotspots of a common amplitude. On each sub-domain of length l = S/K, and translated to ( l,l) to contain exactly one hotspot at x =, the steady-state solution, to leading order, is given by A (x/ǫ) ǫ, if x = O(ǫ); A α, if x = O(1), (.1a) v v v e = ζ [(l x ) l ] [ ] +v, here v = π K q S(γ α) U, (.1b) q +1 u ǫ q 1 ũ e, here ũ e U v q/ KI q and I q Here (y) = sechy is the homoclinic of (.) and ζ (γ α)/(dα ). q dy = q/ 1[Γ(q/)] Γ(q). (.1c) In terms of the criminal and police densities, given by ρ = ǫ va and U = ua q from (1.), e have the folloing: Corollary. Under the same conditions as in Proposition.1, (.1) yields to leading-order that A (x/ǫ) ǫ v, if x = O(ǫ); A α, if O(ǫ) x < l, (.15a) ρ [(x/ǫ)], if x = O(ǫ); ρ ǫ v e α, if O(ǫ) x < l, (.15b) U U [(x/ǫ)] q, if x = O(ǫ); U ǫ q 1 α qu v q/, if O(ǫ) x < l, (.15c) ǫki q KI q here v e and v are given in (.1) and (y) = sechy Figure 1: The steady-state to-hotspot solution for S = 6, γ =, α = 1, U =, ǫ =., D =., D p , for the copson-the-dots q = patrol. Plot is the steady-state solution, closely approximated by the asymptotic result (.15), hich corresponds to the left panel of Fig. 11 belo. In Fig. 1 e use (.15) to plot a to-hotspot steady-state solution for a particular parameter set. This plot clearly shos the concentration behavior of A, ρ, and U near the hotspot locations. From (.15), e observe that the criminal density near a hotspot is independent of the total police deployment U and patrol focus q. Since q > 1, the police density U(x) is small in the outer region, but is asymptotically large near a hotspot. We observe that our leading-order asymptotic result in (.15) for the hotspot steady-state is equivalent to simply replacing U in Proposition.1 and Corollary. of [] ith qu /(q +1). Since q/(q +1) > 1 for q > 1, e conclude that, for the same parameter values and level U of total police deployment, the steady-state hotspot amplitude is smaller for the RD model (1.) ith predator-prey type police interaction than for the RD model of [] ith simple police interaction. 6

7 The NLEP for a K-Hotspot Steady-State Pattern To analyze the linear stability of a K-hotspot steady-state solution, e must extend the singular perturbation approach used in [] to derive the corresponding nonlocal eigenvalue problem (NLEP). This is done by first deriving the NLEP for a one-hotspot solution on the reference domain x l, subject to Floquet-type boundary conditions imposed at x = ±l. In terms of this canonical problem, the NLEP for the finite-domain problem < x < S ith Neumann conditions at x =,S is then readily recovered as in [] (see also [11]). Since the analysis to derive the NLEP is similar to that in [], e only outline it belo. Further details on the derivation of the NLEP are given in Appendix A..1 Linearization ith Floquet Boundary Conditions Let (A e,v e,u e ) denote the steady-state ith a single hotspot centered at x = in x l. We introduce the perturbation A = A e +e λt φ, v = v e +e λt ǫψ, u = u e +e λt ǫ q η, (.1) here the asymptotic orders of the perturbations (O(1), O(ǫ) and O(ǫ q )) are chosen so that φ, ψ, and η are all O(1) in the inner region. By substituting (.1) into (1.) and linearizing, e obtain that ǫ φ xx φ+ǫ v e A eφ+ǫ A eψ = λφ, (.a) D ( A e v ex φ+ǫa ) eψ x x ǫ A ev e φ ǫ A eψ ǫ (+q)u e v e A q+1 e φ ǫ +q v e A +q e η ǫ u e A +q e ψ = λǫ ( A e v e φ+ǫa eψ ), (.b) D ( qae q 1 u ex φ+ǫ q A q ) eη x x ǫ τλ ( qae q 1 u e φ+ǫ q A q eη ). (.c) As in [], for K, e impose Floquet-type boundary conditions at x = ±l for ψ, η, and φ: η(l) η( l) η x (l) η x ( l) ψ(l) = z ψ( l), ψ x (l) = z ψ x ( l). (.) φ(l) φ( l) φ x (l) φ x ( l) Here z is a complex-valued parameter. Since φ is localized near the core of the hotspot, it is only the Floquet-type boundary condition for the long-range components η and ψ that is essential to the analysis belo. We ill consider the case of a single hotspot, here K = 1, separately in. belo. For K, the NLEP associated ith a K-hotspot pattern on [ l,(k 1)l] ith periodic boundary conditions, on a domain of length Kl, is obtained by setting z K = 1, hich yields z j = e πij/k for j =,...,K 1. For these values of z j in (.) e obtain the spectral problem for the linear stability of a K-hotspot solution on a domain of length Kl ith periodic boundary conditions. To relate the spectra of the periodic problem to the Neumann problem, in such a ay that the Neumann problem is still posed on a domain of length S, e proceed as in of [] (see also of [11] and the appendix of []). There it as shon that e need only replace K ith K in the definition of z j. In this ay, the Floquet parameters in (.) for a hotspot steady-state on a domain of length S = lk, having K interior hotspots and Neumann boundary conditions at x = and x = S is z = z j e πij/k for j =,...,K 1. For these values of z, the folloing identity is needed belo: (z 1) ( ) πj = Re(z) 1 = cos 1, z K j =,...,K 1. (.) We no begin our derivation of the NLEP. For (.a), in the inner region here A e ǫ 1 / v, v e v, e have that ψ ψ() ψ. It follos that the leading-order term Φ(y) = φ(ǫy) in the inner expansion of φ satisfies Φ Φ+ Φ+ ψ() = λφ. (.5) 7 v /

8 In the outer region ǫ x l, to leading order e obtain from (.) that φ ǫ α ψ/[λ+1 ǫ α v e ] = O(ǫ ), ψ xx, η xx. (.6) The goal of the calculation belo is to determine ψ(), hich from (.5) yields the NLEP. To do so, e must derive appropriate jump conditions for ψ x and η x across the hotspot region centered at x =. This calculation, summarized in Appendix A, then leads to linear BVP problems for ψ and η, from hich e can calculate ψ(). As shon in Appendix A, e obtain that the outer approximation for ψ(x) satisfies ψ xx =, x l; e [ψ x ] = e 1 ψ()+e η()+e, ψ(l) = zψ( l), ψ x (l) = zψ x ( l), (.7a) here e have defined [a] a( + ) a( ). Defining (...) (...) dy, the coefficients e j, for j =,...,, are e Dα, e ũe q+, v / v 1+q/ e 1 v q/ +q, e Φ+ ũ e v (q 1)/ (q +) q+1 Φ. (.7b) This BVP (.7) is defined in terms of η(), hich must be found from a separate BVP (see Appendix A). To achieve a distinguished balance in this BVP e introduce ˆτ = O(1) by ˆτ ǫ q τ, so that the police diffusivity D p ǫ D/τ becomes D p ǫ 1 q D/ˆτ, here ˆτ ǫ q τ. (.8) In terms of ˆτ, e determine η() from the folloing BVP, as derived in Appendix A: η xx =, x l; d [η x ] = d 1 η()+d, η(l) = zη( l), η x (l) = zη x ( l). (.9a) In terms of v and ũ e, as defined in (.1), the constants d, d 1, and d, are defined by d Dα q, d 1 ˆτλ q, d v q/ ˆτλqũ e q 1 Φ. v (q 1)/ To calculate ψ() and η(), e need the folloing simple result, as proved in Lemma.1 of []: (.9b) Lemma.1 (Lemma.1 of []) On x < l, suppose that y(x) satisfies y xx =, l < x < l; f [y x ] = f 1 y()+f ; y(l) = zy( l), y x (l) = zy x ( l), (.1) here f, f 1 and f, are nonzero constants, and let z satisfy (.). Then, y() is given by [ ] f (z 1) 1 f y() = f f 1 =. (.11) l z f [1 cos(πj/k)]/l+f 1 Lemma.1 ith f = e, f 1 = e 1, and f = e η()+e, yields ψ() from (.7). Similarly, η() is found from (.9) by using Lemma.1 ith f = d, f 1 = d 1, and f = d. In this ay, e get e η()+e ψ() =, and η() =. (.1) e [1 cos(πj/k)]/l+e 1 d [1 cos(πj/k)]/l+d 1 Upon combining these to results, and using (.7b) and (.9b) for e and d, respectively, e determine ψ() as [ ] 1 e d ψ() = D j α e +e 1 D j α q, (.1) +d 1 8 d

9 here e have defined D j, hich satisfies D j < D j+1 for any j =,...,K, by D j D [ ( )] πj 1 cos, j =,...,K 1, here l = S l K K. (.1) To determine the coefficient ψ()/v / in (.5) in terms of the original parameters, hich ill yield the NLEP, e next need to simplify the expressions for e 1, e, e, d 1, and d in (.7b) and (.9b), by using (.6) for ũ e and an explicit formula for the integral ratio q+ / q, as given in (.9). A short calculation yields that qu q+ e 1 = +, e v / =, e (q +1)Kv v q/ = Φ+ U v q+1 Φ (q +), (.15a) K q ( q ) U v q 1 Φ d 1 = ˆτλ, d = ˆτλq. (.15b) K q v q/ Upon substituting (.15) into (.1), e obtain, after some algebra, that = χ j ( ( Φ )+χ 1j (q +) here e have defined χ j Here κ q is defined by ψ() v / 1 q+1 ( Φ )+χ q+ j q 1+κ q +v / D j α /, χ 1j χ j κ q, χ j χ j ( q 1 ) Φ, (.16a) q ˆτλκ q ˆτλ+D j α q v q/ / q ). (.16b) κ q U v K q+ = qu q [ω(q +1)], here ω S(γ α) qu q +1. (.17) In calculating κ q above, e evaluated the integral ratio in (.17) using (.9) and then recalled (.1) for v. From (.16b), e first derive the NLEP for the mode j =, hich corresponds to synchronous perturbations of the hotspot amplitudes. For this mode, e have D = from (.1). Therefore, from (.16b), the coefficients reduce to χ = 1/(1+κ q ), χ 1 = κ q /(1+κ q ), and χ = κ q /(1+κ q ). With these values, e substitute (.16a) into (.5) to obtain the folloing NLEP for the synchronous mode: Proposition. For ǫ, K, q > 1, < U < U,max = (q +1)S(γ α)/(q), D = ǫ D = O(1), and τ O(ε ), the linear stability on an O(1) time-scale of a K-hotspot steady-state solution for (1.) to synchronous perturbations of the hotspot amplitudes is determined by the spectrum of the folloing NLEP for Φ(y) L (R): Φ q+1 +κ Φ q(q +) κ q+ qq [ L Φ 1+κ q q 1 ] Φ = λφ, (.18) q here L Φ Φ Φ+ Φ and κ q is defined by κ q = qu /[ω(q +1)], here ω S(γ α) qu /(q +1). The NLEP (.18) for the synchronous mode depends only on κ q, and is independent of the criminal and police diffusivities. Remark. In. e sho that the NLEP in (.18) also governs the linear stability of a one-hotspot steady-state solution. 9

10 Next e consider the asynchronous modes here j = 1,...,K 1. For these modes, in order to obtain an NLEP ith as fe bifurcation parameters as possible, e introduce in (.16b) to additional rescaled parameters D u and τ u defined by D j = α D u, ˆτ = D j α q v q/ τ q u. (.19) v / By using (.19) in (.16), an NLEP is obtained by substituting (.16a) into (.5). The result is summarized as follos. Proposition. For ǫ, K, q > 1, < U < U,max = (q +1)S(γ α)/(q), D = ǫ D = O(1), and τ O(ε ), the linear stability on an O(1) time-scale of a K-hotspot steady-state solution for (1.) for the asynchronous modes j = 1,...,K 1 is characterized by the spectrum of the folloing NLEP for Φ(y) L (R): L Φ χ ( ( Φ ) χ 1 q+1 ( Φ (q +) ) χ q+ q 1 ) Φ q = λφ, (.a) q here L Φ Φ Φ+ Φ and = sechy is the homoclinic of (.). Here the coefficients of the multipliers are χ = 1 τ u λ, χ 1 = χ κ q, χ = χ κ q 1+κ q +D u 1+τ u λ ; κ q = q U q +1 ω, q ω S(γ α) q +1 U. (.b) For a given q > 1, the spectrum of the NLEP (.) depends on the three key parameters D u, τ u, and κ q. To relate these parameters to the original criminal diffusivity D e use (.19) and (.1), and then (.1) for v to get ( ) S D = α v / K [ 1 cos ( ω S )]D πj u = K K π α [ 1 cos ( )]D πj u, j = 1,...,K 1, (.1) K here ω is defined in (.b). In addition, to map τ u to the original police diffusivity D p, e simply substitute (.19) for ˆτ into (.8) for D p and use (.1) for D j and (.1) for v. In this ay, e obtain D p = Sǫ 1 q q Kα q[ 1 cos ( πj K )] ( ) q ( ) ω 1 K, j = 1,...,K 1. (.) τ u We refer to the NLEP (.) as a universal NLEP, since e need only determine, ith respect to the bifurcation parameters D u, τ u and κ q, hen all discrete eigenvalues of (.) satisfy Re(λ). The regions of linear stability ith respect to these key parameters can then be mapped to corresponding regions of stability ith respect to the original parameters D and D p (for a given U, S, γ, α, and q) by using (.1) and (.). Correspondingly, e ill also identify parameter ranges for hich the NLEP predicts instabilities oing to it having a discrete eigenvalue in Re(λ) >.. Derivation of the NLEP for a Single Hotspot: K = 1 case To derive an NLEP for the case of a single hotspot, e simply impose Neumann boundary conditions directly at x = ±l in (.7) and (.9). This yields that ψ(x) = ψ() and η(x) = η() on x l. From (.9) and (.7), e conclude that η() = d ψ() = 1 (e η()+e ) = 1 ( e e ) d. d 1 e 1 e 1 d 1 By using the explicit expressions for the coefficients given in (.15), e calculate ψ()/v /, hich leads to the NLEP from (.5). In this ay, e obtain that the NLEP for a single hotspot is also given by (.18) of Proposition.. 1

11 No Unstable Eigenvalues for the NLEP (.18) for the Synchronous Mode In this section, e study the NLEP (.18) of Proposition., hich applies to either amplitude perturbations of a onehotspot steady-state or synchronous perturbations of the amplitudes of a multi-hotspot steady-state..1 Numerical Computations We first sho numerically that (.18) has no unstable eigenvalues for any κ q and q > 1. To do so, e rite (.18) as L Φ (a here the constants a, b, and c are defined by a = q+1 Φ+b Φ+c ) q 1 Φ = λφ, (.1) κ q (q +) (1+κ q ) q+, b = (1+κ q ), c = qκ q (1+κ q ) q. (.) To numerically compute the discrete eigenvalues of (.1) e convert this NLEP into a linear algebra problem using finite differences. As e are interested only in even solutions, e consider (.1) on [, ]. Since (y) decays exponentially as y +, e truncate the positive half-line to the large interval x [,L], here e chose L = (decreasing L to 1 changes the results belo by less than.1%). We discretize Φ(x j ) Φ j here x j = j x, for j =...N 1 and x = L/(N 1), ith N = 1. Increasing N to changed the results belo by less than 1%. We use standard centered differences to approximate Φ and the Trapezoid rule to approximate integrals in (.1). In this ay, e obtain the matrix eigenvalue problem MΦ = λφ, here Φ (Φ 1,...,Φ N ) T. The eigenvalue of M ith the largest real part then provides an excellent approximation to the principal eigenvalue of (.1) Figure : Numerical approximation of the principal eigenvalue of (.1) for several values of q and ith κ (,1). We observe that λ < in all cases, and that the principal eigenvalue is rather insensitive to changes in q. In terms of κ q this numerical approximation of the principal eigenvalue of (.1) is plotted for q = 1, q =, and q = in Fig.. The results shon in Fig. suggests that (.18) has no unstable eigenvalues for any κ q and q 1. Although the NLEP (.18) is only relevant to the stability of a hotspot steady-state only hen q > 1, as a partial confirmation of the numerical results in Fig. e no sho ho to determine λ analytically from (.1) hen q = 1. Let Φ and λ be any eigenpair of (.1) for hich Φ and Φ. We multiply (.1) by and integrate. By using the identity L =, e obtain (λ ) Φ = (a+b) 5 Φ c 5 Φ. (.) 11

12 Next, e integrate (.1) upon recalling L Φ = Φ Φ+ Φ. This yields (λ+1) Φ = Φ [(a+b) Φ+c ] Φ. (.) By eliminating Φ and Φ from (.) and (.), e then obtain the folloing quadratic equation for λ: c ( (a+b) 5 ( )+ λ +(a+b) )(c 5 ) +λ+1 =. (.5) For q = 1, e obtain from (.) that a + b = /, and c = κ 1 / [ (1+κ 1 ) ]. Upon substituting these expressions into (.5), and using / = 1 and 5 / = /, e get ( λ+ 1 )( λ+ ) =. (.6) 1+κ 1 Since κ 1, e conclude that the principal eigenvalue of (.1) hen q = 1 is λ = 1 1+κ 1. (.7) Setting κ 1 = 1 gives λ = 1/, hich agrees ith the numerical result shon in Fig. arising from a discretization of (.1).. Hybrid Analytical-Numerical Approach q We no give an alternative approach that provides a sufficient condition to ensure that the NLEP (.18) has no unstable eigenvalues. This sufficient condition is then investigated numerically. For this hybrid analytical-numerical approach, e rite the NLEP (.18) in the alternative form L Φ f()φ = λφ, (.8a) here L Φ Φ Φ+ Φ, and f() is defined by f() (1+κ q ) + (q +)κ q (1+κ q ) q+1 qκ q q+ (1+κ q ) q 1. q (.8b) When κ q =, here f() = /, the NLEP (.8a) has no unstable eigenvalues by Theorem 1 of [6] (see also Lemma. of [11]). We multiply (.8a) by the conjugate Φ and integrate over the real line. Upon integrating by parts and taking the real part e get I q [Φ R ]+I q [Φ I ] = λ R Φ, (.9) here Φ = Φ R +iφ I and λ = λ R +iλ I. Here the quadratic form is defined by I q [Φ] (Φ ) +Φ Φ Φ f()φ +. (.1) To sho that λ R <, so that there are no unstable eigenvalues of the NLEP (.8), it is sufficient to sho that the quadratic form I q [Φ] is positive definite. In Appendix C e establish the folloing lemma for I [Φ]. Lemma.1 We have I [Φ] > Φ. 1

13 Since I [Φ] >, our strategy is to continue in κ q > until e reach a point for hich I q [Φ] ceases to be positive definite. To analyze this transition, e observe that I q [Φ] = ΦLΦ, here LΦ is the linear operator f()φ LΦ L Φ Φ f(). (.11) Since L is self-adjoint, it follos that I q [Φ] is positive definite if and only if L has only negative eigenvalues. This motivates the consideration of the folloing zero-eigenvalue problem for L: To analyze (.1) e use (.11) to get LΦ =, Φ L (R), Φ. (.1) Φ = f()φ L 1 Φ + L 1 f(). (.1) Define c 1 = f()φ and c = Φ. By multiplying (.1) by f() and then by e get the linear system c 1 = c 1 f()l 1 +c f()l 1 f(), c = c 1 L 1 L 1 +c f(). (.1) Upon using L 1 = /, and integrating by parts, e obtain that (.1) has a nontrivial solution iff g(κ q ) =, here f() g(κ q ) det f()l 1 1 f() ( ) f() f() = 1 1 ( ) f()l 1 f(). (.15) When κ q =, e have f() = /. Upon using L 1 = /, = 16/, and = π, e calculate g() = <. (.16) π Thus, hen κ q =, the only solution to (.1) is c 1 = c =, and so (.11) becomes L Φ =, hich has no nontrivial even solution. By increasing κ q, e conclude that a sufficient condition for guaranteeing no unstable eigenvalues of the NLEP (.8) is that on the range < κ q < κ q e have g(κ q ) <. Here κ q is defined by κ q = sup{κ q g(t) <, t (,κ q )}. (.17) In Fig. e plot g(κ q ) versus κ q for q =,,. These results ere obtained by numerically evaluating the integrals in (.15), after computing L 1 f() from a BVP solver. On the range for hich g(κ q) <, e conclude that I q [Φ] is positive definite so that the NLEP (.8) has no unstable eigenvalues. As a partial confirmation of the results in Fig. e no sho ho to calculate g(κ q ) analytically hen q =. When q =, e have that ψ L 1 f satisfies here e, e 1, and e, are defined by L ψ = f, f = e +e 1 +e, (.18a) e = (1+κ ), e 1 = κ 1+κ, e = κ 1+κ. (.18b) By using (.18a) for f(), e calculate f() 1 = (1+κ ) + κ (1+κ ) 1 = (1+κ ) (1+κ ). (.19) 1

14 Figure : Plot of numerically computed g(κ q) versus κ q, as defined in (.15), for q =,,. For q =, g(κ ) is given analytically in (.). On the range of κ q for hich g(κ q) <, the NLEP (.8) has no unstable eigenvalues. Next, upon using L =, L = and L (+y ) =, e calculate from (.18a) that ψ L 1 f = e + e 1 + e (+y ). (.) Upon using (.) and (.18b), e obtain after some rather lengthy, but straightforard, algebra that ( ( ) fl 1 f = κ (1+κ ) 1 ( ) κ (1+κ ) +κ + [ ] κ 5 8(1+κ ) ) + (1+κ ) +. (.1) We then simplify the expression in (.1) by using = 16/, =, = π and 5 = π/. In this ay, and by combining the resulting expression ith (.19), e obtain from (.15) that [ 1 g(κ ) = 16(1+κ ) (κ +1) 8 ( κ +5κ + )] π. (.) Recallingthedefinition ofthethresholdκ in(.17), asimplecalculation using(.) yields κ = 1 [7+ ] 6+56/π 7.7. The formula for g(κ ) in (.), and the threshold κ, agrees ith the numerical results shon in Fig.. In summary, e have shon that henever g(κ q ) < in (.15), the NLEP (.8) has no unstable eigenvalues. This sufficient condition for stability as implemented numerically for q, and analytically for q =, hich shoed that the NLEP has no unstable eigenvalues for κ q belo some threshold. On the other hand, the numerical results in Fig. obtained from a finite-difference approximation suggested that the NLEP (.8) has no unstable eigenvalues for all κ q >...1 No Unstable Eigenvalues for q = and any κ In this subsection e provide a different approach to prove that for q = that there are no instabilities associated ith synchronous perturbations of the hotspot amplitudes for any κ >. For q =, this NLEP has the general form ] L Φ [a Φ+b Φ+c Φ = λφ, Φ L (R); L Φ Φ Φ+ Φ. (.) We ill convert this NLEP into one ith a single nonlocal term proportional to Φ by using the to identities (cf. []): L =, L ( ) = ( ). (.) Let Φ and λ be any eigenpair of (.). We first multiply (.) by, and then use the first of (.), together ith Green s identity, to obtain L Φ = ΦL = Φ = a Φ+b Φ+c Φ+λ Φ. (.5a) 1

15 Next, e multiply (.) by, and then use the second of (.), together ith Green s identity, to obtain L Φ = ΦL = Φ = a 5 Φ+b 5 Φ+c 5 Φ+λ Φ. (.5b) Equations (.5a) and (.5b) provide a matrix system for Φ and Φ of the form ( λ b 5 a )( ) ( 5 b a Φ c ) = 5 Φ ( λ+c ) Φ. (.6) By inverting the matrix in (.6), e obtain that (c+aλ) 5 Φ = ( λ) ( a ) +b 5 Φ, Φ = ( λ+c ) ( λ)+bλ 5 ( λ) ( a ) +b 5 Φ, (.7) provided that ( ( λ) a ) +b 5. (.8) By substituting (.7) into (.), and using =, e obtain after some algebra the folloing NLEP ith a single nonlocal term: L Φ γ Φ = λφ, here γ = ( λ)(aλ+c) (λ ) ( a ) b 5. (.9) Conversely, suppose that Φ and λ is any eigenpair of (.9). Upon multiplying (.9) by, and then by, and using (.), e obtain from (.9) that ( λ) 5 Φ = γ Φ, ( ) Φ = γ +λ Φ. (.) Next, by adding and subtracting terms in (.9) e get ] L Φ [a Φ+b Φ+c Φ+ξ = λφ, ξ ( γ c ) Φ b Φ a Φ, (.1) hich reduces to (.) only hen ξ =. We solve (.) for Φ, and for Φ hich requires λ. Then, upon using (.9) for γ, e can readily verify from (.1) that ξ =. Therefore, any eigenpair of (.9) ith λ is also an eigenpair of (.). For q =, the coefficients a, b, and c in (.) associated ith synchronous perturbations of the hotspot amplitudes are a = κ (1+κ), b = (1+κ), c = κ (1+κ), (.) here e label κ κ. By combining (.) and (.9), and using / = / and 5 / = /, e get γ = hile the condition (.8) becomes (λ )(κ 1) 9/. κ( λ)(λ ) [(λ )(κ 1) 9], (.) To prove that (.9), ith γ as in (.), has no unstable eigenvalues e ill use a key inequality that can readily be derived by proceeding as in (.) of [7] (see also equation (.7) in of [5]). Suppose that (.9) has an eigenvalue ith Re(λ). Then, the folloing inequality must hold: ( ) T γ 1 +Re [ λ(γ ] 1), (.) 15

16 here the bars denote modulus. From (.), e calculate that γ 1 = λ κ+λ(9κ+)+6 (κ 1)λ 1κ 6. (.5) We ill no use (.), ith (.5), to sho that the NLEP (.9) cannot have any purely complex eigenvalues of the form λ = iω. For λ = iω, e rite γ 1 in (.5) as γ 1 = z 1 z, z 1 = 6+ω κ+iω(9κ+), z = 6(1+κ)+iω(κ 1). (.6) Using / = /, e calculate from (.) that T = 8 z 1 ( z +Re iω z ) 1 z = 8 z 1 z Re ( ) iωz1 z z = 1 [ 8 z1 z +ωim(z 1 z ) ]. (.7) Upon substituting (.6) into (.7), e obtain after some rather lengthy, but straightforard, algebra that 1 T = [κ(κ+1)ω 6(1+κ) +16ω (κ 1) + ω ( 16κ +κ+6 ) ] +8. (.8) 18 This shos that T > holds κ > and ω. From our key inequality (.), it follos that the NLEP (.9) does not undergo a Hopf bifurcation for any κ. To conclude the analysis of linear stability e use a continuation argument in κ. With a, b, and c as given in (.), the NLEP (.) has no unstable eigenvalues hen κ = by Theorem 1 of [6] (see also Lemma. of [11]). By our established correspondence beteen the to NLEPs (.) and (.9), this linear stability result can also be seen from (.9), as (.9) has no unstable eigenvalues ith λ hen κ =. This latter result is immediate since hen κ =, e have γ = in (.9). Therefore, (.9) reduces to L Φ = λφ, hich has no unstable eigenvalues ith λ (cf. [7]). Next, if e continue in κ, e claim that all eigenvalues of (.) must remain in the stable left half-plane Re(λ). We establish this by contradiction. Suppose that at some point, κ = κ >, an eigenvalue crosses the imaginary axis, i.e. it satisfies Re(λ) =. At this point κ, and for this purely complex eigenvalue, it follos that λ(k 1) 9/ and so the restriction (.8) holds. Therefore, this eigenvalue must satisfy the NLEP (.9) ith only one nonlocal term. Our proof above that the NLEP (.9) has no purely imaginary eigenvalue provides the required contradiction. 5 Analysis of the NLEP: Competition Instability In this section e ill analyze zero-eigenvalue crossings for the NLEP (.), corresponding to asynchronous perturbations of the hotspot amplitudes. This zero-eigenvalue crossing ill yield K 1 critical values of the criminal diffusivity D. We ill determine the behavior of this stability threshold in terms of the police focus parameter q and policing level U. By using L =, and noting that χ = hen λ =, it follos that (.) has a zero eigenvalue hen = χ +(q +)χ 1, (5.1) for hich Φ =. By using (.b) for χ and χ 1, e solve (5.1) for D u to conclude that the NLEP (.) has a zero eigenvalue at the critical value of D u given by D u = 1 (1+qκ q), here κ q = q U q +1 ω. (5.) Then, by using (.1), it follos that a zero-eigenvalue crossing occurs at D = D j, for j = 1,...,K 1, given by [ ] S D j = 8K π α [ 1 cos ( )] ω + q U πj q +1 ω, j = 1,...,K 1. (5.) K 16

17 The smallest such threshold D c = min j D j on j = 1,...,K 1, referred to as the competition stability threshold, occurs hen j = K 1. We rite D c as D c D K 1 = S 8K π α [1+cos(π/K)] g(u ;q), here g(u ;q) is defined on the range U < U,max = S(γ α)(q +1)/(q) by ( ) q g(u ;q) ω + q +1 U ω = (1 q)ω +qs(γ α)ω, here ω = S(γ α) qu /(q +1). (5.b) (5.a) For a general value of q > 1, oing to the presence of the three distinct nonlocal terms in (.), it is analytically intractable to perform a full linear stability analysis of hotspot steady-states on either side of the zero-eigenvalue crossing value D = D c. For the specific q = cops-on-the-dots case, here some key identities can be used to reduce (.) to an NLEP ith only one nonlocal term, this linear stability problem is studied in 6 by using a hybrid analytical-numerical approach. Hoever, for a general q > 1, in 5.1 e sho analytically that the NLEP (.) alays has a unique unstable eigenvalue in Re(λ) > henever D > D c and τ u =, and has no unstable eigenvalue hen D < D c. This partial result proves that, for an infinite police diffusivity, the hotspot steady-state is alays unstable for any q > 1 hen D exceeds D c. In the remainder of this sub-section, e examine ho the competition stability threshold D c depends on the degree q of patrol focus and the level U of police deployment. From (.1), the maximum A max of the steady-state attractiveness field is A max ǫ 1 ω/(kπ), hich decreases as either ω decreases or as K increases. From Corollary., e observe that the criminal density ρ at the hotspot locations is ρ max = [()] =, hich is independent of q and U, ith ρ = O(ǫ ) aay from the hotspot regions. As a result, the total crime is reduced primarily by decreasing the number of stable steady-state hotspots on the given domain. As such, e seek to tune the police parameters q and U so that the range of diffusivity D for hich a K-hotspot steady-state is unstable hen τ u = (see 5.1 belo) is as large as possible. This corresponds to minimizing the competition stability threshold D c in (5.), hich is determined in terms of g(u ;q) in (5.b). We first fix q > 1, and study ho g(u ;q) depends on U. On < U < U,max S(γ α)(q +1)/(q), e find that dg = ω q [ ] 6q(q 1) du q +1 q +1 U +( q)s(γ α). (5.5) This shos that dg/du < on < U < U,max henever 1 < q <. Thus, hen the patrol is not too focused, i.e. hen 1 < q <, increasing the overall policing level leads to a larger range of D here the hotspot steady-state is unstable hen τ u =. For q >, (5.5) also yields that ( ) dg q > on < U < S(γ α) du (q 1) < U,max ; ( ) dg q < on S(γ α) < U < U,max. (5.6) du (q 1) Therefore, ith an overly focused police patrol (i.e. q > ), the hotspot steady-state is destabilized only by having a sufficiently large policing level. This is illustrated in Fig. here e plot g(u ;q) versus U for several values of q. Next, e fix U in < U < U,max and determine ho g(u ;q) depends on q for q > 1. We readily calculate that dg dq = ωu (q +1) [ ω(q +)(q 1) q U ], here ω S(γ α) qu /(q +1). (5.7) This shos that dg/dq < if < ω < q U /[(q +)(q 1)]. By relating ω to U, this inequality yields [ ] 1 dg dq < hen U q,max 1+ < U < U,max. (5.8) (q +)(q 1) The qualitative interpretation of this result is that if the policing level is sufficiently close to its maximum value U,max, an increase in the patrol focus parameter q yields a larger range in D here the hotspot steady-state is unstable hen τ u =. 17

18 Figure : Competition stability threshold nonlinearity g(u ;q), as defined in (5.b), versus U on the range < U < U,max S(γ α)(q +1)/(q) for q = 1.5,.,.5,.,.5, hen S = 6, γ =, and α = 1. Smaller values of q correspond to larger values of U,max. Notice that g is not monotone in U hen q >. From (5.), the competition threshold D c is a positive scaling of g(u ;q). 5.1 Infinite Police Diffusivity: An Instability Result for D > D c When τ u =, e no prove that the NLEP (.), corresponding to asynchronous perturbations of the amplitudes of the steady-state hotspot pattern, has an unstable positive real eigenvalue henever D u > 1 (1+qκ q). This ill establish that a multi-hotspot steady-state is unstable for an infinite police diffusivity henever D exceeds the competition stability threshold D c defined in (5.). When τ u =, e have χ = and so (.) reduces to an NLEP ith to nonlocal terms L Φ χ ( ( Φ ) χ 1 q+1 ) Φ 1 (q +) = λφ, here χ q+ = (1+κ q +D u ), χ 1 = χ κ q. (5.9) To analyze (5.9), e first reformulate it into an NLEP ith only one nonlocal term by using the key identity L ( ) = ( ) (cf. []). Upon multiplying (5.9) by, and then using Green s identity, e readily calculate that ) ( 5 5 ) Φ ( χ λ = χ 1 (q +) q+1 Φ. (5.1) q+ Since 5 / = / from (.5), (5.1) yields that ( χ 1 (q +) ) 5 q+1 Φ Φ = 9χ λ, (5.11) q+ provided that λ 9χ /. Then, by substituting (5.11) back into (5.9), and using χ 1 = χ κ q, e obtain the folloing equivalent NLEP ith only one nonlocal term (provided that λ 9χ /): L Φ χ c (λ) q+1 Φ q+ = λφ, here χ c (λ) χ κ q (q +) ( λ 9χ λ ). (5.1) To interpret the apparent restriction that λ 9χ /, e observe that since χ 1 = χ κ q is proportional to U (see (.b) for the definition of κ q ), it follos from (5.1) that for any eigenpair for hich q+1 Φ for any q > 1, e must have λ = 9χ,j / if and only if U =. For the case of no police, this recovers the result in equation (.17) of [11] for the unique discrete eigenvalue of the linearization of a K-hotspot steady-state of the basic to-component crime model. We no sho that the reformulated NLEP (5.1) has an unstable real eigenvalue henever D u > 1 (1+qκ q). To do so, e convert (5.1) into a root-finding problem. We rite Φ = χ c (L λ) 1 q+1 Φ/ q+, multiply both sides by q+1, and then integrate over the real line. In this ay, and by using (5.9) for χ, e readily find that any discrete 18

19 eigenvalue of (5.1) in Re(λ) > must be a root of ζ(λ) = defined by ζ(λ) C c (λ) F(λ), (5.1a) here C c (λ) 1 χ c (λ) = (1+κ q +D u ) 9 q+1 (L λ) 1 +, and F(λ). (5.1b) κ q (q +) κ q (q +)(λ ) q+ When D u > 1 (1+qκ q), e claim that ζ(λ) = has a real root in < λ <, hich yields an unstable eigenvalue for the NLEP (5.1). To sho this, e use L = to calculate F() = q+1 L 1 ( )/ q+ = 1/, and observe that F(λ) + as λ ν, here ν = is the unique positive eigenvalue of L (cf. [7], []). Moreover, e observe that C c (λ) as λ, and C c () = (κ q +D u 1/) κ q (q +) hich yields that C c () > 1/ hen D u > 1 (1+qκ q). With these properties of C c (λ) and F(λ), it follos from the intermediate value theorem that ζ(λ) has a root at some value of λ on < λ <. This simple result proves that a multi-hotspot steady-state is unstable for τ u =, i.e. for an infinite police diffusivity, henever D exceeds the competition stability threshold D c in (5.). Next, by using a inding number criterion, e no obtain a more precise result for the spectrum of the NLEP (5.1), hich pertains to the special case τ u =. We do so by determining the number N of zeroes of ζ(λ) in Re(λ) >, hich corresponds to the number (counting multiplicity) of unstable eigenvalues of the NLEP (5.1). To determine N, e calculate the inding of ζ(λ) over the Nyquist contour Γ traversed in the counterclockise direction that consists of the positive and negative imaginary axis, defined by Γ + I ( < Im(λ) < ir, Re(λ) = ) and Γ I ( ir < Im(λ) <, Re(λ) = ) respectively, together ith the semi-circle C R defined by λ = R > for arg(λ) < π/. From (5.1b), C c (λ) is a meromorphic function ith a simple pole at λ =, hereas F(λ) is analytic in Re(λ) except at the simple pole at λ =. The simple poles of C(λ) and F(λ) do not cancel as λ, since hen restricted to the real line e have F(λ) + hile C(λ) as λ. Therefore, ζ(λ) = C(λ) F(λ) has a simple pole at λ =. Then, since ζ(λ) is bounded on C R as R, and ζ(λ) = ζ(λ), e let R and obtain from the argument principle that, N = 1+ 1 π [argζ] Γ + I. (5.1) Here [argζ] Γ + denotes the change in the argument of ζ as λ = iλ I is traversed don the positive imaginary axis < λ I <. I To calculate this argument change, e let λ = iλ I and decompose ζ(iλ I ) = ζ R (λ I )+iζ I (λ I ) and F(iλ I ) = F R (λ I )+ if I (λ I ), to obtain from (5.1) that Im[ζ(iλ I )] ζ I (λ I ) = bλ I 9+λ F I (λ I ), I here b 9[κ q (q +)] 1 and F I (λ I ) Im[F(iλ I )] is given by F I (λ I ) = λ I (5.15a) [ ] q+1 1 L +λ I. (5.15b) q+ Proposition. of [] established that F I (λ I ) > on λ I > hen q =. Based on the numerical evidence shon in Fig. 5, for both integer and non-integer values of q, e make the folloing conjecture: Conjecture 5.1 Consider F I (λ I ) Im[F(iλ I )] as defined by (5.15b). Then, F I (λ I ) > on λ I > holds for all q > 1. 19