Modelling Territoriality and Wolf-Deer Interactions

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

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

Salt Lake City, UT, USA 84112, and 2Department of Applied Mathematics, University of Washington, Seattle, WA, USA 98195.

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 scentmarking[?, ?]. 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 Van Ballenberghe et al.[?] with permission from The Wildlife Society (copyright holder).

Figure 2: Winter yards and summer home ranges of radio-tagged deer in relation to wolf-pack territories. Wolf-pack territories: A, Glenmore L. pack; B, Newton L. pack; C, Ensign L. pack; D, Greenstone L. pack; E, Pagami L. pack; F, Harris L. pack; G, Birch L. pack in northeastern Minnesota. Redrawn from Fig. 4 in Hoskinson and Mech[?] with permission. (Roads have been removed and color has been added.)

Figure 3: Territorial patterns arise naturally as stable steady-state solutions to the model. Shown is a stable steady-state solution for (1)– (4) for the case l = 1.0, f = 1.0, m = 1.1, J_{c_{u}} = −uq \frac{(x−x_{u})}{|x−x_{u}|}, J_{c_{v}} = −vp\frac{(x−x_{v})}{|x−x_{v}|}, J_{d_{u}} = −0.3∇u, J_{d_{v}} = −0.3∇v, x_{u} = 0 and x_{v} = 1. A territorial boundary is found at the point x = 0.5. Near this boundary the expected wolf densities, u(x) and v(x), are low and expected RLU densities, p(x) and q(x), are high. Numerical solution used the Method of Lines and Gear’s Method.

Figure 4: Deer are found primarily in the narrow corridors between the