Analysis of a model for wolf territories

M. A. Lewis^{1}  K. A. J. White ^{2}  J. D. Murray^{3}

1Department of Mathematics, University of Utah, JWB 233,

Salt Lake City, UT, USA 84112, and

2Department of Zoology, Downing Street, University of

Cambridge, Cambridge, UK CB2 3EJ and 3Department of Applied Mathematics, University of Washington, Seattle, WA, USA 98195. email: mlewis@math.utah.edu

office phone: (801) 581–6195  FAX: (801) 581–4148

A mechanism for territorial pattern formation in wolves (Canis lupus) is proposed and analysed using a spatially explicit mathematical model incorporating wolf movement and scent-marking. Model results reflect field observations from northeastern Minnesota: buffer zones where wolves are scarce arise between adjacent packs and near these buffer zones there are increased levels of scent-marking^{?,?}. Territories, themselves, are arranged in a striking spatial mosaic (Fig. 1) which covers the wolf range. In the model formulation no assumptions are made about actual existence or extent of wolf territory and buffer zones; these arise naturally as stable steady-state solutions to the equations. We show mathematically how reduced predation by wolves in the buffer zones provides a refuge for prey species. Field studies support this; distribution of a primary prey species, white-tailed deer (Odocoileus virginianus), correlates negatively with that of wolves—deer are found primarily in buffer zones^{?,?} (Fig. 2).

Elaborate spatial patterns of wolf scent-marks^{?} are used to advertise precise information about territorial claims^{?}. Behavioral studies indicate that it is the raised leg urination (RLU) which plays the key role in scent-marking^{?,?}. Out of a typical pack of 5–15 wolves^{?}, a few mature dominant members leave RLUs regularly upon trails; a travelling wolf may encounter one RLU every two to three minutes^{?}. Marking frequency approximately doubles near territorial borders, giving rise to ‘bowl-shaped’ distributions of RLUs across territories, with the raised edges of the olfactory bowl located at the territorial boundaries^{?}. Observations indicate scrutiny of RLUs^{?} and some aversion to foreign RLUs made by neighboring packs^{?}.

We propose a mechanism for generating the territorial patterns using a mechanistic, spatially explicit mathematical model with simple rules for scent-marking and movement. Making no underlying assumptions about the size and extent of the wolf territories themselves, we show how the territorial patterns actually arise naturally as stable steady-state solutions to the equations.

Our key modelling assumption is that wolf movement and behavior is mediated by the absence/presence of foreign RLUs. In the absence of foreign RLUs, (i) movement is primarily dispersive as individuals search for food and other resources, and (ii) RLU-marking occurs at low levels. The presence of foreign RLUs elicits two responses: (i) movement towards an organizing center for the pack such as a den, above-ground rendezvous site or familiar location^{?}; and (ii) increase in RLU-marking frequency^{?}. The RLU marks, themselves, are assumed to lose intensity with age.

The expected location of an RLU-marking wolf is described by a probability density function denoting the chance of finding it at point x and time t. For any given pack, these are summed over the number of RLU-marking wolves to yield the expected density of RLU-marking wolves, henceforth referred to as the expected density of wolves in a pack. For a model of two adjacent interacting wolf packs, pertinent state variables thus are the expected densities of wolves in packs 1 and 2, (u(x, t) and v(x, t)); and of RLUs from packs 1 and 2, (p(x, t) and q(x, t)).

Components of the wolf movement are expressed mathematically by:

\frac{∂u}{∂t} = −∇ · \{J_{d_{u}} + J_{c_{u}}\} ,  (1)

and

\frac{∂v}{∂t} = −∇ · \{J_{d_{v}} + J_{c_{v}}\} ,  (2)

where J_{d_{u}} and J_{d_{v}} describe dispersive movement via diffusion and J_{c_{u}} and J_{c_{v}} describe movement towards organizing centers situated at x_{u} and x_{v} via convection towards x_{u} and x_{v}, respectively.

Specific forms for the fluxes in (1)–(2) were derived under the assumptions that (i) dispersive movement is modelled via simple diffusion (J_{d_{u}} = −d∇u, J_{d_{v}} = −d∇v); and (ii) convective movement is at a rate proportional to the expected density of foreign RLU (J_{c_{u}} = −cuq \frac{(x−x_{u})}{|x−x_{u}|}, J_{c_{v}} = −cvp\frac{(x−x_{v})}{|x−x_{v}|}, c > 0).

Equations for RLU-marking and decay are:

\frac{∂p}{∂t} = u (l + mq) − fp,  (3)

and

\frac{∂q}{∂t} = v (l + mp) − fq,  (4)

where l > 0 denotes low level continual RLU-marking, m > 0 describes increased RLU-marking levels in the presence of foreign RLUs, and f > 0 represents first order RLUs decay kinetics.

Analysis indicates the existence of a stable, spatially heterogeneous steadystate solution to (1)–(4). Describing territorial patterns and corresponding levels of the spatially-distributed RLUs, this solution (Fig. 3) can be found analytically by integrating steady-state versions of (1)–(4) subject to zeroflux boundary conditions for u and v. The solution predicts the partitioning of available space into territories and the formation of territorial boundaries near which there are low levels of expected wolf density and high levels of expected RLU density (Fig. 3). This clearly reflects the field observations regarding territorial patterns, interpack buffer zones and bowl-shaped RLU densities. Thus the simple rules for scent-marking and movement given here are sufficient to explain territorial patterns.

Here, scent-marks are crucial to the formation of territories; if scentmarking is not included (l, m = 0 in (3)–(4)) then p, q → 0 and equations (1) and (2) are reduced to diffusion equations so that the territories do not form. It can also be shown that a necessary condition for the bowl-shaped RLU patterns with interior maximums for p and q (Fig. 3) is that foreign RLUs elicit a sufficiently large increase in the RLU-marking rate. Mathematically, this condition is met when m exceeds a critical value.

We now consider an extension of the above model which can be used to predict the striking negative correlation between wolf territories and the white-tailed deer distribution (Fig. 2). For the sake of illustration we assume:

(i) wolf territories are stationary and stable (Fig. 3); (ii) Holling Type I functional response of deer to wolf predation; (iii) Beverton-Holt densitydependent population dynamics. Denoting h(x, t) as the expected density of deer, we have

\frac{∂h}{∂t} = −ψ(u(x) + v(x))h,  (5)

where u(x) and v(x) denote the expected densities of pack 1 and pack 2 wolves, ψ is the predation rate and 0 ≤ t ≤ T, the year being T time units long. Steady-state solutions, yielding constant values when deer densities are compared from one spring to the next, satisfy the Beverton-Holt dynamics:

h(0, x) = \frac{λh(T, x)}{1 + (λ − 1)h(T, x)/K} ,  (6)

where λ > 1 and K measure the population growth rate and carrying capacity of the deer in the absence of predation. Solving (5) subject to the condition (6) gives steady-state solutions as

h(0, x) = max{0, K (λ − exp{[u(x) + v(x)]ψT }) /(λ − 1)}.  (7)

Thus deer are found primarily where the combined wolf densities are lowest — in the buffer zones between packs (Fig. 4).

As noted by Mech^{?}, evidence for a similar situation involving deer and human societies is given by Hickerson’s study of Sioux and Chippewa Indians from 1780–1850, “The Virginia deer and intertribal buffer zones in the upper Mississippi Valley”^{?}. Warfare between the two tribes gave rise to a buffer zone which was normally unoccupied. In turn, reduced hunting pressure in this buffer zone effectively provided a refuge for the deer. However, when a long truce was maintained, the buffer zone was destroyed and famine ensued.

List of Figures

Figure 1: Radiolocations and home range boundaries of 5 adult and yearling timber wolves radiotracked via aircraft, 27 May – 22 October 1971 in northeastern Minnesota. Redrawn from Fig. 4 in VanBallenberghe:WM-43-1 with permission from The Wildlife Society (copyright holder).

Figure 2: (a) Scent mark densities p(u(x), v(x)) and q(u(x), v(x)) are determined uniquely by (??)–(??) m_{u}(q) and m_{u}(q) are convex functions. (b) If m_{u}(q) and m_{u}(q) exhibit switching then p(u(x), v(x)) and q(u(x), v(x)) may not be determined uniquely if u(x) and v(x) are sufficiently small.

Figure 3: (a) Three regions where solutions to equation for wolf pack 1 (??) must lie. These are given by (??)–(??). (b) The maximum and minimum values for u(0). The area under each curve integrates to unity and thus satisfies (??).

Figure 4: Functional forms for the movement behavior function c(q) and the marking behavior function m(q): (a) convex c(q), (b) convex m(q),

(c) switching in c(q) and (d) switching in m(q).

Figure 5: Time-independent solution to (1)–(2), (??)–(??), (??)–(??) with m_{u} = m_{v} = 0 and c_{u}(q) = γq, c_{v}(p) = γp, γ = 1, (Figure 4(a)) and d_{u} = d_{v} = 0.1. Shown is the large-time numerical solution with the expected density of wolves from pack 1 (u(x)) and the expected density of wolves from pack 2 (v(x)). This corresponds to the analytical solution to the system of logistic equations (??). Note that u(x)+v(x) = 2 pointwise across the domain so there is no buffer zone. The expected scent mark densities are identical to the expected wolf densities. Numerical solution used the Method of Lines and Gear’s Method.

Figure 6: Time-independent solution to (1)–(2), (??)–(??), (??)–(??) with c_{u}(p) = γp, c_{v}(q) = γq (Figure 4(a)) and m_{u}(q) = μq, m_{v}(p) = μp, (Figure 4(b)) and γ = 1, d_{u} = d_{v} = 0.333, μ = 1.1. (a) Shown is the large-time numerical solution with the expected density of wolves from pack 1 (u(x)), the expected density of wolves from pack 2 (v(x)), the expected scent mark density from pack 1 (p(x)) and the expected scent mark density from pack 2 (q(x)). This corresponds to the analytical solution to equation (??) and the similar equation for u. Note the bowl shaped scent mark densities. Intersection of the line u = 1/μ with u(x) yields the location of the maximum value for p(x) which corresponds to the edge of the bowl (??). (b) The corresponding cumulative expected wolf density (u(x) + v(x)) and cumulative scent mark density (p(x) + q(x)). The function (1 + μu)(1 + μv) is constant across the domain as predicted by (??). Note the the elevated scent mark density and buffer zone at x = 0.5. Numerical solution used the Method of Lines and Gear’s Method.

Figure 7: Analytical solution to the territoriality ODEs (??)–(??) subject to the integral constraints (??) with c(q) = c_{∞}H(q − q_{c}) (Figure 4(c)) and m(q) = μq (Figure 4(b)). The solution for u(x) and v(x) is given by (??)–(??) with u(0) and x_{c} determined by (??)–(??). The corresponding solutions for p(x) and q(x) are given by (??)–(??). Note the width of the buffer zone.

Figure 8: Stable time-independent equilibria for (??)–(??) with m(q) = m_{∞}H(q − q_{m}). Region A: (p, q) = (u, v), region B: (p, q) = (u, (1 + m_{∞})v), region C: (p, q) = ((1 + m_{∞})u, v) and region D: (p, q) = ((1 + m_{∞})u, (1 + m_{∞})v). Note that if q_{m}/(1 + m_{∞}) < u, v < q_{m} then the equilibrium cannot be determined uniquely, being either case A or B above. If parameters are chosen so that k = u(0) + (1 + m_{∞})v(0) − m_{∞}q_{m} > q_{m} (??) then the indeterminate region can bypassed in phase space. A sample trajectory is shown with three legs of the trajectory as given in (??).

Figure 9: An analytical solution to the territoriality ODEs (??)–(??) subject to the integral constraints (??) with m(q) = c_{∞}H(q − q_{m}) (Figure 4(d)) and c(q) = γq (Figure 4(a)). The corresponding solution trajectory is given in Figure 8. The solution method is the same as described in Section 6.1 except with the relationship between u(x) and v(x) in each subregion given by (??). Note the abrupt jumps in p(x) and q(x).

Figure 10: Two dimensional territories. Each surface describes a group of wolves and their scent marks. The height of the surface describes the expected density of wolves and the color of the surface describes the expected scent mark density. The shading follows colors of the rainbow with red being low and violet being high. Solutions were generated by solving (1)–(2), (??)–(??), (??)–(??) numerically using finite differences and an alternating direction implicit method until the a stationary solution was effectively reached. The domain size is [0, 1] × [0, 1] and the mesh size is 40×40. The movement function is given by

c(q, r) = \frac{c_{∞}}{2} (1 + tanh(γ(q − q_{c}))) tanh(αr)

and the scent marking function is given by

m(q) = \frac{m_{∞}}{2} (1 + tanh(μ(q − q_{m}))) .

In the case with three interacting packs, an additional wolf movement and scent marking equation is added and the functions m and c are modified to depend on the cumulative foreign scent mark density. (a) Two interacting packs with dens at opposite corners of the domain (d = 0.3, c = 3, α = 100, γ = 2, m_{∞} = 1.5, c_{∞} = 1, μ = 10, q_{c} = 1, q_{m} = 1 and the color scale ranges from 0 to 4). (b) The cumulative expected wolf density and expected scent mark density surface for (a) above. (c) Three interacting packs with dens in two adjacent corners and at the midpoint of the opposite side (d = 0.3, c = 3, α = 100, γ = 2, m_{∞} = 1.5, c_{∞} = 1, μ = 7, q_{c} = 0.75, q_{m} = 1 and the color scale ranges from 0 to 5). (d) The cumulative expected wolf density and expected scent mark density surface for (c) above.

Acknowledgements

This work was supported in part by US National Science Foundation Grants DMS–9222533 and DMS–9457816 and an Alfred P. Sloan Research Fellowship (MAL) and by US National Science Foundation Grant DMS–9296848 (KAJW, JDM).