Home Range Formation in Wolves Due to Scent Marking

Transcription

1 Blletin of Mathematical Biology () 64, doi:1.16/blm.1.73 Available online at on Home Range Formation in Wolves De to Scent Marking BRIAN K. BRISCOE, MARK A. LEWIS AND STEPHEN E. PARRISH Department of Mathematics, University of Utah, Salt Lake City, UT 8411, U.S.A. s: Social carnivores, sch as wolves and coyotes, have distinct and well-defined home ranges. Dring the formation of these home ranges scent marks provide important ces regarding the se of space by familiar and foreign packs. Previos models for territorial pattern formation have reqired a den site as the organizational center arond which the territory is formed. However, well-defined wolf home ranges have been known to form in the absence of a den site, and even in the absence of srronding packs. To date, the qantitative models have failed to describe a mechanism for sch a process. In this paper we propose a mechanism. It involves interaction between scent marking and movement behavior in response to familiar scent marks. We show that the model yields distinct home ranges by this new means, and that the spatial profile of these home ranges is different from those arising from the territorial interactions with den sites. c Society for Mathematical Biology 1. INTRODUCTION Field biologists have observed wolves and coyotes sing scent marks to identify home ranges and territories (defended home ranges) (Mech, 1991). Canids can distingish between a foreign scent mark and one of their own (Peters, 1974; Merti- Millhollen et al., 1986) and, when encontering a foreign scent mark, may mark over it (Peters and Mech, 1975). Foreign scent marks may also elicit an avoidance response, with individals exhibiting an increased probability of moving towards the center of their home range (Peters and Mech, 1975). Continm territory models, based on the above elements of behavior, have been proposed by Lewis and Mrray (1993) and others (Lewis et al., 1997; White et al., 1996, 1998; Moorcroft et al., 1999), and can be derived from random walks, as shown in Moorcroft et al. (1999). Together the models address how the interaction between scent marking and movement behavior in response to foreign scent marks can lead to territorial Athor to whom correspondence shold be addressed. Department of Mathematical and Statistical Sciences, LAB 545 B, University of Alberta, Edmonton, Alberta T6G G1, Canada. 9-84// $35./ c Society for Mathematical Biology

2 6 B. K. Briscoe et al. patterns. In these models a den site acts as a focal point for the movement of individals and this focal point is a necessary element of the model if territories are to form (White et al., 1996). However, well-defined wolf home ranges have been known to form in the absence of a den site (Rothman and Mech, 1979), and even in the absence of srronding packs (Mech, 1991). To date, the qantitative models have failed to describe a mechanism for this process. In this paper we propose a mechanism for the process. It involves interaction between scent marking and movement behavior in response to familiar scent marks. We show that the model yields distinct home ranges by this new means, and that the spatial profile of these home ranges is distinct from those arising from the territorial interactions with den sites present. Or modeling approach is to describe the change in the location of an individal via a random walk process. This leads to a copled partial differential eqation (PDE)/ordinary differential eqation (ODE) model throgh the limiting processes of the random walk. The modeling approach is otlined in Aronson (1985) and has been applied widely (Okbo, 198). Two applications relevant to this paper inclde modeling of aggregating insect poplations by Trchin (1989), and the modeling of aggregating poplations with long-range interactions between individals by Lewis (1994). In or model formlation it is assmed that the pack is initially isolated, with no neighboring packs, and no established den site. Individals both prodce scent marks and modify their movement behavior in response to their pack s scent marks. The model, described as a random walk process, assmes that the transition probabilities describing spatial movement depend pon the local level of scent marks, and that scent mark prodction also depends pon existing scent marks. Analysis of the copled ODE/PDE system shows spontaneos formation of welldefined home ranges for reasonable parameter ranges. These home ranges have abrpt edges and are stable to pertrbations. Energy methods and nmerical methods are sed to calclate the home range patterns. The reslts in this paper contrast with previos model reslts where foreign scent marks and a den site are necessary ingredients for the process of territorial pattern formation (Lewis and Mrray, 1993; White et al., 1996; Moorcroft et al., 1999).. MODEL DERIVATION We first consider a derivation in one spatial dimension, and then we extend this to a two-dimensional (D) model..1. One-dimensional approach. A classical approach to deriving a PDE model, based on the movement of an individal comes from a random walk on the real line (Okbo, 198). In or formlation, we assme that an individal s movement behavior is modlated by local scent density individals preferentially remain in

3 Home Range Formation 63 areas with high scent levels. We begin the random walk derivation by discretizing both time and space into small intervals of length δt and δx, respectively. We assme that each individal at each point in time decides whether or not to move, and if so, in which direction. We also assme that an individal can only move a distance of δx, or remain in its crrent position in each sccessive time step, δt. We let L(x, t), R(x, t) and N(x, t) be the probabilities of moving to the left, to the right, and of not moving, respectively, over the next time step. Taking these as the only possibilities, we have that N(x, t) + L(x, t) + R(x, t) = 1. (1) If scent level is high, N(x, t) is close to 1, and the wolf tends not to move, however if scent levels are low, N(x, t) is small, and the individal is more likely to move. If we make the assmption that a wolf moves depending only on the scent level at its present location, it is logical to take L(x, t) = R(x, t). Combining this with (1) we get 1 N(x, t) L(x, t) = R(x, t) =. () The forward Kolmogorov, or Fokker Planck eqation, relates these probabilities to the following PDE: t = D [(1 N(x, t))(x, t)] (3) x where (x, t) is the expected poplation density. Details of the derivation are given in Appendix A. However we note here that D = lim δx,δt δx /δt is the diffsion coefficient, which is derived nder the assmption of rapid movement and freqent switches in movement direction. For simplicity, we can make the sbstittion D(x, t) = D (1 N(x, t)), (4) which brings s to t = [D(x, t)(x, t)]. (5) x Now, trning or attention to the scent marking eqation, we make some simple, biologically relevant assmptions. We define p(x, t) to be the density of scent marks. First of all, we assme that wolves mark with scent at a constant rate, γ, bt increase the marking rate when they enconter scent marks other than their own. The increasing fnction m( p) describes enhanced scent mark rates in the presence of existing scent marks. Finally, it is assmed that scent marks decay with time at a constant rate, µ. The eqation for scent is then p t = (x, t)(γ + m(p(x, t))) µp(x, t). (6)

4 64 B. K. Briscoe et al. m(p) Types of fnctions for m(p) p m(p) 1 Scent Density 6 p Figre 1. This shows the two possible scent-response fnctions, m( p), that we se. The top figre is the piecewise linear, and the bottom is the piecewise qadratic, fond in Appendix B. The first form we consider for m(p) gives the simplest type of response, linear increase p to some level of p(x, t), at which the response becomes constant, as in the first graph of Fig. 1. The second form is similar, bt it is piecewise qadratic, as in the second graph of Fig. 1. We assme that the probability of not moving, N(x, t), increases monotonically with scent density p(x, t). In the absence of detailed data on the precise relationship between N(x, t) and p(x, t) we take this to be N(x, t) = p(x, t) α + p(x, t). (7) The parameter α describes the sensitivity of movement behavior to scent mark levels when scent mark levels are low. Using this and eqation (4), we can find an explicit expression for D(x, t) D(x, t) = D 1 + p(x, t)/α. (8) Or system of eqations is now flly defined as ( ) t = D, (9) x 1 + p/α p t = (γ + m(p)) µp. (1)

5 Home Range Formation 65 In or analyses we will assme a domain < x < L from which the wolves do not leave and hence zero-flx bondary conditions, are appropriate. ( ) D = at x =, L, (11) x 1 + p/α.. Two spatial dimensions. As mch insight as a one-dimensional model can give s, it falls short of reality. In their natral environment, wolves move over two dimensions. Or D derivation ses a similar approach to the one in the previos section. Once again, we begin with a random walk, bt this time we add another degree of freedom. As before, we let L(x, t), R(x, t) and N(x, t) be the probabilities of moving to the left, to the right and not moving, respectively. We then add to these the possibilities of moving forward and backward, which we will denote with F(x, t) and B(x, t). It is also important to note that these vales now depend on two spatial parameters, and therefore x = (x 1, x ). Taking these as the only possibilities, we have N(x, t) + R(x, t) + L(x, t) + F(x, t) + B(x, t) = 1. (1) If we assme a wolf s decision to move only depends on his crrent position, we can also assme that R(x, t) = L(x, t) = F(x, t) = B(x, t) = 1 N(x, t). (13) 4 A derivation analogos to the one in the previos section yields a system of eqations similar to the one-dimensional (1D) case. ( ) t = D 1 + p/α p t (14) = (γ + m(p)) µp, (15) on the soltion domain (Parrish, 1998) where now D = lim δx,δt δx /4δt. Zero flx bondary conditions become ( ) D n = (16) 1 + p/α on the bondary δ, where n is the otwardly-oriented nit normal vector.

6 66 B. K. Briscoe et al. 3. A NONLOCAL APPROXIMATION In the previos eqation for the scent, we assmed that the level of an individal s response depends only on local scent densities. This is not realistic however, becase wolves can indeed sense scent markings at a distance (Peters, 1974) We now take into consideration how scent markings that are near the position of an individal wolf play a role in the wolf s decision-making. Let k(x) be a symmetric, olfactory kernel which integrates to nity and defines the locally averaged scent mark density by k(z x)p(z, t) dz. (17) This can be approximated by p(x, t) = p(x, t) + σ p(x, t) (18) x where σ is the second moment of the olfactory kernel k, a measre of its variance or spread (Mrray, 1989). When decisions are made based on the locally averaged scent mark density, p can replace p in eqation (9) and in the fnction m in eqation (1). To keep or analysis analytically tractable, we only consider the case where m( p) is sed in eqation (1), with σ. This change leads s to a new scent marking eqation: p t = (γ + m(p(x, t))) µp. (19) We will show that σ > is reqired to ensre that (9), (1) are well posed Two-dimensional approximation. In two spatial dimensions, the analysis yields an analogos reslt, with p(x, t) = p(x, t) + σ p(x, t). () Sbstitting this into (15), we have an eqation identical to (19) where x is now a D spatial coordinate. It shold be noted that in the limit σ = the kernel k corresponds to a delta distribtion centered at the wolf s crrent position, so that only local scent densities are taken into consideration. In this case (19) is simplified to (1) or (15).

7 Home Range Formation ANALYSIS OF EQUATIONS 4.1. Nondimensionalization. Before doing any analysis, we nondimensionalize (9) and (19) in order to redce the nmber of parameters. It is possible to redce the nmber of parameters by for since there are for variables:, p, t and x. The nondimensional variables are ˆx = x L, ˆt = µ ɛ t, ɛ = D µl, û = γ αµ, ˆp = p α, ˆσ = σ L. After making the aforementioned sbstittions and dropping the carets, the nondimensional eqations are with bondary conditions ( ) t =, (1) x 1 + p ɛ p t = (γ + m(α p)) p, () γ ( ) = at x =, 1. (3) x 1 + p The parameter ɛ is small when the domain is large relative to spreading de to local diffsion over the characteristic time scale µ 1. An analogos D nondimensionalization can be sed for (14), (19) to yield on with bondary condition ( ) t =, (4) 1 + p ɛ p t = (γ + m(α p)) p, (5) γ ( ) n = on. (6) 1 + p Note that the total nmber of wolves U is conserved becase ( ) ( ) d dx = dx = n ds =. (7) dt 1 + p δ 1 + p

8 68 B. K. Briscoe et al. 4.. Steady state soltions. We now focs on analysis of the 1D spatial system, (1) (3), while noting that analysis of the analogos D system, (4) (6), is similar. Steady state soltions to (1), () satisfy ( ) =, (8) x 1 + p (γ + m(α p)) p =. (9) γ Application of the zero flx bondary conditions (3) yields 1 + p = const. If the steady state soltion is spatially homogeneos or if σ =, then (γ + m(αp)) p =. γ Solving this eqation for p = f () yields = const (3) 1 + f () (Fig. ). Details are given in Appendix C. We observe that, depending pon the vale of the constant, there will be one, two or three vales of satisfying (3). We define φ() = C, (31) 1 + f () where C is chosen to be the niqe constant that ensres 3 1 φ() d =, (3) and 1 and 3 are the left and rightmost roots of φ as shown in Fig. 3. This shift in (31) by a constant will aid the sbseqent graphical analysis. At steady state, (8) becomes (φ()) =, (33) x which has the soltion φ() = λ (constant). (34) A steady state soltion s, satisfying (34) for some λ, has a corresponding scent mark level p s = f ( s ).

9 Home Range Formation 69 1+f() f().4. 4 (x,t) Poplation Density Figre. These are the fnctions /(1 + f ()) associated with m(p) in Fig. 1. The eqations for these fnctions are fond in Appendix C f().3.5 * 3 * 1 * A 1 A Figre 3. This is the top fnction from Fig., which is simply /(1 + f ()), along with three different constants as in (3). The middle constant is the niqe vale given in (31), C =.99, which satisfies (3), i.e., the constant which makes A 1 = A. The vales of 1, and 3 are.495, 1.1 and.86, respectively.

10 7 B. K. Briscoe et al Dispersion relation. To analyze the stability of a spatially homogeneos steady state soltion ( s, p s ) we linearize, with = s + v and p = p s + q, where v, q 1. Sbstittion into (1), () yields the linearized system ( δv δt vxx ɛ δq δt ( 1 + p s sq xx (1 + p s ) ( m (p s ) s q + σ γ γ m (p s )q xx ), (35) ) ) ( q + v 1 + m(p ) s). (36) γ If we consider soltions of the form ( ) ( ) v(x, t) ŵ = = e st+ikx v, (37) q(x, t) q then sbstitte ŵ into (35) and (36), we have sŵ = ( k s k ( 1+p s ) ( (1+p s ) m(p s) 1 s m (p s ) ɛ γ ɛ γ (1 σ k ) 1 ) ) ŵ = Aŵ. (38) This is the eigenvale eqation for the matrix A. For the eqation to be satisfied for a nontrivial vale, we reqire det(a s I ) =. This gives a characteristic polynomial, which can be solved for s in terms of the wavenmber k. We will also allow σ to vary to see what effect it has on pattern formation. By looking at (37), we observe that the wavenmbers k that correspond to s > will grow with wavelength π/k. Figre 4 shows the dispersion relation for ( s, p s ) = (, f ( )) = (1.1, 3.4) for the cases where σ = and σ =.1. Note that this s is fond on the descending (middle) branch of φ [i.e., φ ( ) < ]. [See Section 4., eqation (C1) in Appendix C, and Fig. 3.] Figre 5 shows the same dispersion relation for σ.1. For the case σ =, the dispersion relation increases monotonically, and hence the smallest wavelength pertrbations grow the fastest. This is the hallmark of an ill-posed problem. From a modeling perspective, we observe that this comes from the random walk process when all scent mark information, however close by, is ignored, except the scent density immediately nder the individal wolf. This problem is corrected if there is any spatial averaging of scent mark information (σ > ). In this case, s > over a range of wavenmbers indicates that spatial patterns may form spontaneosly from the spatially homogeneos soltion. 5. AN ENERGY METHOD As mentioned in Section 4., eqation (34) can have either one, two or three soltions, depending on the vale of λ. This raises the qestion as to whether

11 Home Range Formation 71.4 Dispersion Relation S σ =. σ =.1 4 k 8 Figre 4. This is the dispersion relation when σ =,.1. The first is a bonded, increasing fnction for k >. Becase of this, the high wavenmbers (small wavelengths) grow the fastest. The second increases ntil it attains a maximm vale, and then decreases and becomes negative. In this case, there is a finite range of wavenmbers that will grow. s k σ.1 Figre 5. This is the dispersion relation plotted against σ and k for the system we have been analyzing along with the plane s =. Recall for s >, the area which is above the plane s =, pertrbations of the corresponding wavelength will grow.

12 7 B. K. Briscoe et al. Figre 6. The crve plotted here is where the dispersion relation is zero. The vales of σ and k in the shaded region correspond to growing pertrbations, whereas the vales above the crve lead to decaying pertrbations. If we choose a particlar vale of σ, we can now find the range of wavelengths that will grow. See, also, Figs 4 and 5. spatially heterogeneos discontinos soltions may exist for (1), () in the limit as σ, where = 1 on measre L 1, = on measre L and = 3 on measre L 3 = 1 L 1 L. We investigate this qestion sing an energy method and focs on the case where the domain size is large (ɛ ). Using this method we will show that, providing the total nmber of wolves U is sfficiently large, the lowest energy soltion is L = and = 1 on an interval of length L 1 and = 3 on an interval 1 L 1, where L 1 is fond by solving 1 L 1 + (1 L 1 ) 3 = U. The region over which = 3 is the home range. See, for example, Fig Defining the energy. We define or energy to be a qantity which is bonded below, and show that it is a decreasing fnction of time. The first thing we might wish to do is consider the case where ɛ 1. This is reasonable becase, in or nondimensionalization, ɛ = D /(µl ). In realistic cases, L can be chosen so that L D and µ is of order 1. This redces (1), () to t = (φ()), (39) x where φ is defined in (31).

13 We define or energy to be Home Range Formation 73 E() = 1 where F () = φ(), and observe that Ė() = 1 F((x, t)) dx, (4) F () 1 (x) dx = F ()[φ()] xx dx. (41) t Integrating (41) by parts and sing the bondary condition (11) to eliminate the bondary term yields 1 Ė() = x (F ()) [φ()] dx = x 1 ( ) x [φ()] dx. (4) As long as the flx, φ(), is nonzero, E() will decrease with time. Also, since x φ() is a bonded fnction, the energy, E() is bonded below for. This prompts s to look for a global minimm for the energy, (4). 5.. Minimizing the energy. To minimize E(), we will se Lagrange mltipliers and constraint (7). We define L() = 1 F() dx λ 1 ( U ) dx, (43) which is the qantity to be minimized. If we assme that minimizes this eqation, and + δη(x) is a variation, then a minimm of L( + δν(x)) shold occr when δ =. Taking the derivative with respect to δ we get L δ = 1 evalating this at δ = we get L δ = L δ = (F ( + δη(x))η(x) λη(x)) dx (44) 1 1 (F ()η(x) λη(x)) dx (45) (F () λ)η(x) dx. (46) Since η(x) is an arbitrary fnction, the minimm energy occrs precisely when F () = λ which, when sbstitting F () = φ(), gives s the same condition as (34): φ() = λ. From eqation (4) we observe that φ() = λ implies that E

14 74 B. K. Briscoe et al. φ () λ + λ 1 3 Figre 7. This a plot of φ() as defined in (31) and plotted in Fig. 3. As mentioned, (34) can hold for any constant. The solid line is λ =. The vales 1,, and 3 correspond to φ() = λ. With both constants shown, the soltion to (34) has three soltions, however, it is also possible to find constants sch that (34) has one or two soltions as well (Fig. ). is no longer decreasing (Ė = ). We now consider which vale of λ will give a global minimm for the energy (4). We consider the cases shown in Fig. 7, where three distinct vales of satisfy (34), and observe that (4) becomes E() = N L i F( i ), (47) i=1 where N L i = 1, (48) i=1 and N is the nmber of distinct soltions to (34). Using (47) and (48) we find that the energy eqation redces to a convex combination of F( i ). When N = 3, we have that E() = L 1 F( 1 ) + L F( ) + (1 L 1 L )F( 3 ) (49) We have one more constraint given in (7) which is U = N L i i = U, (5) i=1 where U is the normalized total nmber of wolves. For the case N = 3, the choice of L 1 and L to give the minimm vale for E() (49) has been analyzed in detail by Parrish (1998). Here we give a simple graphical argment. First we observe

15 Home Range Formation 75 F() * 1 * * 3 Figre 8. This is the fnction F() defined in Section 5.1. As given in Fig. 3, the vales of 1, and 3 are.495, 1.1 and.86, respectively. that the fnction F corresponding to φ (Fig. 3), where F () = φ(), has the form shown in Fig. 8. Now for any L 1 and L ( L 1 + L 1) the total poplation nmber U (5) and the energy E (49) are depicted graphically as the coordinates of a point in the triangle whose vertices are [ 1, F( 1 )], [, F( )] and [ 3, F( 3 )] (Figs 9 11). In Fig. 9, the points 1, and 3 are chosen so that φ( 1 ) = φ( ) = φ( 3 ) = λ + > (Fig. 7, pper line), and hence F ( 1 ) = F ( ) = F ( 3 ) = λ + > (Fig. 9). The constraint U = U (5) is depicted by the vertical line segment in Fig. 9, and the minimm possible energy, given that U = U, (E min ) comes at the point where the vertical line segment toches the base of the triangle. The cases λ < and λ = are shown in Figs 1 and 11. Ths the lowest possible energy comes from the case λ = where 1 = 1, = and 3 = 3 (compare Figs 9, 1 and 11). For this case, 1 =.495, = 1.1 and 3 =.86. Using (48) and (5) we find that L 1 =.787 and L 3 =.13. Or analysis has shown that when σ, ɛ, the energy of the system will decrease ntil φ() = λ(const). Each vale of λ defines a 3 vale (within home range density) and a 1 vale (otside home range density) and the low-energy soltion has measre L 1 otside of the home range = 1 and measre 1 L 1 within the home range = 3. The vale of λ which gives the global energy minimm (minimm over all possible λ) is λ =, with corresponding within home range density and otside home range density given by 3 and 1, respectively. We now se nmerical simlation to see whether the global energy minimm is achieved. 6. NUMERICAL SOLUTIONS OF THE SYSTEM Nmerical simlations sed a finite difference, Crank Nicolson method with variable weighting between forward and backward differencing in time for each eqation. For comptational simplicity, eqation () was solved sing forward

16 76 B. K. Briscoe et al. F() F( ) F( 3 ) F( 1 ). (U,E). (U, E min ) 1 U 3 Figre 9. This represents the fnction F(), plotted against. The triangle is the convex combination of F( 1 ), F( ) and F( 3 ), where λ = λ + > and an arbitrary energy vale is shown. The energy associated with a particlar U is somewhere along the vertical line, and is clearly minimized at the bottom of the vertical line. This implies that at the minimm energy level, steady state soltions consist only of the vales of 1 and 3, and is nstable. F() F( ) F( 3 ). F( 1 ) (U, E min ) 1 U 3 Figre 1. As in Fig. 9, bt with λ = λ <. F() F( ) F( ) 1,F() 3. (U, E min ) 1 U 3 Figre 11. As in Fig. 9, bt with λ =. If yo compare the previos cases (Figs 9 and 1), it is clear that the minimm vale of E( ) for any occrs when the bottom line of the triangle connects the local minimms of the fnction.

17 Home Range Formation 77 Figre 1. Beginning with a groped distribtion of wolves, a distinct region of high density marked by an abrpt change to lower densities emerges. The initial distribtion of wolves was given by 1 cos(π x). In this figre, the steady state vales are 1 =.49, 3 =.79 which [from (3)] corresponds to /(1 + f ()) =.98 which is within.3% of the minimm energy constant C in (31). In this figre, ɛ =.1. Figre 13. Starting with the same initial conditions as Fig. 1, we set σ =.1 which increases the range over which an individal can detect scent markings. The transition from low to higher densities is not so dramatic, which is consistent with the dispersion relation (see Fig. 4 and discssion in Section 6). In this figre, ɛ =.1.

18 78 B. K. Briscoe et al. Figre 14. At higher densities, the soltion moves toward a niform distribtion, = 1. In this figre U = 1. and σ =.1. Figre 15. At lower densities, the soltion very rapidly moves toward a niform distribtion, = 1. In this figre U =.1 and σ =.1.

19 Home Range Formation 79 Figre 16. We begin here with an approximate niform distribtion with random pertrbations, and σ =. Since pertrbations of infinitely small wavelengths grow the fastest, there are many regions of high density separated by areas of low density. This simlation was exected with 1 spatial grid points. Or analysis sggests that as the nmber of grid points approaches, these areas of higher density wold become infinitely small. Simlations were also done with p to 1 grid points, and match the analytical reslts. For this particlar case, 3 =.84 and 1 =.494 which corresponds well with 3 =.86 and 1 =.495. Figre 17. Starting with the same initial conditions as Fig. 16, we set σ =.1. Jst as in Fig. 17, the difference is that transitions can no longer be so abrpt, and hence the compted soltion does not have infinitely many areas of high density. It does, however, develop into two distinct areas of higher density. With different random pertrbations, the soltion can exhibit as many as for different areas of high density.

20 8 B. K. Briscoe et al. Figre 18. Starting with different random pertrbations, this soltion can exhibit for different areas of high density. differencing (i.e., explicitly) and eqations (1), (3) were solved sing backward differencing (i.e., implicitly). In all simlations shown, we sed 1 grid points in the x direction, which reslts in only very small differences from the same simlations done with 8 spatial grid points. Figres 1 18 were generated by down-sampling the amont of points sed in both space and time. It is evident from Figs 8 and 11 that the global energy minimm (λ = and = 1 = 1 on measre L 1, = 3 = 3 on measre 1 L 1) can only be achieved when the normalized total nmber of wolves, U, lies between 1 and 3. We chose parameter vales as given in Appendix B, so that 1 =.495 and 3 =.86, as shown in Fig. 3. Except where noted, the vale of ɛ was ɛ =.1. The initial wolf density was given by 1 cos(π x), with the normalized total nmber of wolves eqal to U = 1.. The scent density was initially zero. The cases σ = and.1 are shown in Figs 1 and 13, respectively. In Fig. 1, reslting steady state vales 1 =.49 and 3 =.79 correlate closely with the 1 and 3 vales given above. The differences are probably attribtable to the fact that the global minimm energy analysis is valid only in the limit ɛ =. We find that for nonzero initial conditions for scent density, the minimm energy level was not always achieved, althogh, eqation (34) is always satisfied. This is a topic for frther research. The edge of the territory in Fig. 1, evidenced by the abrpt edge on the nmerical simlation, is consistent with the dispersion relation shown in Fig. 5. When σ is nonzero the edges on the simlation are ronded off and the 1 and 3 vales are modified (Fig. 13). When the normalized total wolf density exceeds 3, the entire region becomes territory (Fig. 14) and when the normalized wolf density is below 1 no territories

21 Home Range Formation 81 form (Fig. 15). For these cases it can be shown that the lowest energy soltions are spatially niform soltions with = 3 > 1, and = 1 < 1, respectively. When initial data are distribted randomly abot a spatially niform wolf density, the σ = case gives an ill-behaved soltion (Fig. 16), consistent with or analysis of Section 4.3. However, when σ =.1 the soltion is no longer ill-behaved, bt the location of the territory (or territories) depends pon the initial data (Figs 17 and 18). Althogh sch randomly distribted initial data are not as biologically relevant as the groped initial data that give rise to a single territory (Figs 1 and 13), we inclde these additional simlations to nmerically illstrate the fll range of model behavior. 7. DISCUSSION We have shown that scent marking alone is sfficient to form a home range if there are two ingredients: (1) a propensity to remain near existing scent marks and () positive feedback, with increased scent marking in the presence of familiar scent marks. Each of these assmptions is spported by biological literatre for wolves (see Introdction). Reslting home ranges are flat-topped with abrpt edges. This model is most appropriate for sexally immatre wolves that are in minimal contact with other packs. The model in this paper contrasts with previos models which rely on pack interactions and existing den sites for territorial pattern formation. These other models modlate movement by the level of foreign scent marks, as opposed to familiar scent marks, and reslting territorial patterns show expected density gradally decreasing with distance from the den site (Lewis et al., 1997). When there is no spatial averaging of scent information (σ = ), or model is illposed, bt once a small level of spatial averaging is inclded (σ > ), the problem becomes well-posed. An analogos sitation can be seen for an aggregation model proposed by Trchin (1989) [see also the discssion in Lewis (1994)]. Indeed or redced model, given by eqation (39), is Trchin s insect aggregation model, althogh derived by a different means. Ths the energy methods in this paper can be applied to models of aggregation more general than the formation of home ranges throgh scent marking. Or model is, by necessity, simplistic once home ranges are formed it is well known that cognitive maps are sed by canids to navigate arond familiar areas (Peters, 1979), and we have not inclded this aspect. Frthermore, we assme that territorial scent marks are made by a single individal, or a single grop of individals moving together, sch as an alpha pair. However, the modeling and analysis can be considered as a first step towards realistic spatially explicit models which inclde interactions with familiar scent marks.

22 8 B. K. Briscoe et al. ACKNOWLEDGEMENTS This work was spported, in part, by a National Science Fondation grant DMS (MAL) and by the Canada Research Chairs Program. Special thanks go to Marks Owen of Loghborogh University, U.K., and David Eyre of the University of Utah who were kind enogh to assist in the early development of this project. Comments from Pal Moorcroft and Thomas Hillen helped improve the manscript. APPENDIX A: FORWARD KOLMOGOROV EQUATION We wish to derive the Fokker Planck, or forward Kolmogorov eqations from the assmptions in Section. We define q(x, t) to be the probability density fnction of a wolf s location at time t. By the assmption that an individal can move at most a distance δx at each time step, we have that q(x, t) = R(x δx, t δt)q(x δx, t δt) +N(x, t δt)q(x, t δt) +L(x + δx, t δt)q(x + δx, t δt). What this says is that in order to be at point x at time t then an individal at time t δt was either at x δx and moved to the right, x + δx and moved to the left, or at x, and did not move dring that time step. In doing this, we have also sed the assmption that probability of moving is independent of everything besides local scent density. Using (), we can rewrite q(x, t) in terms of N(x, t) as q(x, t) = 1 N(x δx, t δt) q(x δx, t δt) + N(x, t δt)q(x, t δt) 1 N(x + δx, t δt) q(x + δx, t δt). Now, in order to simplify this, we do a power series expansion of all terms involving either δt or δx. After doing this, and ptting all terms that inclde δt on the left-hand side of the eqal sign, we are left with ( δx )( ) δtq t = [(1 N(x, t))q(x, t)] + h.o.t. (A1) x We take the limit as δx and δt in sch a way that lim lim δx δx δt δt = D.

23 Home Range Formation 83 This is called the diffsion limit, and has been commonly sed to derive diffsionrelated eqations. Another advantage in doing this is that the higher order terms in the eqation all go to zero. If we now define (x, t) = q(x, t) to be the expected wolf density, what we are now left with is which is (3). t = x [D (1 N(x, t))(x, t)] (A) APPENDIX B: THE FUNCTION m(p) { ap p b m(p) = a b p b a ap m(p) = a b p + 5a p 9b 16 b p b a 3b 4a p 5b 4a p 5b 4a (B1). (B) In the simlations, we took the following vales for varios parameters: γ = 1 µ = 1 α = 1 a =.5 b = D = 1. APPENDIX C: THE FUNCTIONS f () AND φ() We define f () and φ() for each particlar case of m(p) as follows: for the piecewise linear m(p) (B1) { 1+ f () = (C1) { ( ) C 4 φ() = C 4 3. (C) For the piecewise qadratic m(p) (B) f () = (C3)

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