Despite the complexity of wolf and deer behaviours and ecology, the stability of the pack territories and wolf-deer distribution observed in northeastern Minnesota suggests that there may be basic mechanisms underlying the spatial structure and dynamics of the ecosystem. The background details provided in the last section form the basis for the modelling we now discuss. The principal modelling motivation is whether or not simple behavioral rules can help elucidate the following questions (not all of which we address here).
(i) Can we show how pack territories form, determine their size and explain why they are stable for many years?
(ii) When deer, as prey, are included can we show why they are found mainly in the buffer zones between pack territories?
(iii) With seasonal changes can we explain winter increase in buffer zone trespass, wolf-wolf altercation, wolf starvation and territory change?
(iv) Can we predict wolf dynamics with low winter deer populations?
(v) Can we quantify our predictions of population dynamics, territory and buffer zone sizes and seasonal changes, based on behavioral parameters? How sensitive are these predictions to behavioral changes?
(vi) Do buffer zones stabilize wolf-deer interactions by providing a refuge for the deer, and if so, does a refuge act to dampen population oscillations or prevent extinction or both?
(viii) Do biannual migrations act as a stabilizing factor in wolf-deer interactions?
As mentioned there are seasonal changes in wolf ecology. Since we are primarily interested here in territory formation we consider the formation and maintenance of territories during the summer months and so we do not include yearly birth processes. Consequently, the models focus on wolf movement patterns, which in later sections we couple to deer mortality caused by wolf predation, and aspects of the deer movement.
Due to the small numbers of both species there are potentially significant periods of time during which areas of territory are not occupied by a wolf (or deer). In view of this, it makes sense to use a probabilistic approach in which state variables are taken to be expected densities of wolves at a point x and time t; direct field observations typically will not yield the exact densities.
In view of the probabilistic approach and the choice of RLU marking as the method of territory delineation, a two wolf pack model could include the following state variables.
u(x, t) = expected density of wolves from pack number 1 v(x, t) = expected density of wolves from pack number 2 p(x, t) = expected density of RLUs from wolf pack number 1 q (x, t) = expected density of RLUs from wolf pack number 2.
During the summer months, pack members focus their movements around the den but they must necessarily spend time away from the den foraging for food. At the simplest level, we anticipate that wolf movement, independent of responses to other wolf packs, is dominated by (i) dispersal as the wolves search for food and other activites (like RLU marking) and (ii) movement back towards the den as the wolves return to the social organizing centre, the den, to care for the pups. So, a typical word equation for a single pack without RLU and deer input with this scenario is
Rate of change in expected wolf density
= Rate of change due to movement of wolves towards the den
+ Rate of change due to dispersal of wolves away from high density regions in search of food.
The key question is how to model the spatial movements.
Field studies indicate that wolves use cognitive maps and are aware of their relative locations within the territory. Consequently, movement back towards the den site tends to be more or less in a straight line. Mathematically, such movement can be represented by directed motion, or convection, with a flux, Ju which takes the form, for the u-pack,
where xu denotes the location of the den and cu (x — xu) is the space-dependent velocity of movement; Okubo (1986) used a similar form in his model for insect dispersal that we discussed in Chapter 11, Volume I in which we used the discontinuous function cu (x—xu) = cu sgn (x—xu) where cu, the speed of movement, is constant. A continuous version of (14.1) which describes slowing and eventual stopping as wolves approach the den site is cu (x — xu) = cu tanh(jr)x—u, (14.2)
r where r = || x — xu ||. The parameter cu now measures the maximum speed of the wolf when moving towards the den and j measures the change in the rate of convective movement as the den is approached. In the limit as j ^ to (14.2) approaches the discontinuous form. In the presence of foreign RLUs the coefficient describing the speed of movement may be modified to include a response to the foreign RLU marking as described later.
Let us now consider movement due to foraging activity. In the first case we assume a plentiful and homogeneous food supply and in the second, discussed below, the deer density is explicitly incorporated into the model.
In the first case, the simplest assumption is that there is no preferred direction of motion for foraging and so is a random walk process as could occur if the food supply were uniformly distributed throughout the region. An extension to this assumes that movement may be density-dependent. As we now know, mathematically such movement can be represented by a diffusion flux, Ju, which for the wolf pack u is
where D(u) = duun, with constant du and n > 0, is the density-dependent diffusion coefficient. For n positive, density-dependence can be interpreted as an increased rate of movement in regions which are more familiar to the wolf pack.
Consider the simplest scenario of a single isolated wolf pack. Combining movement back to the den to care for the young, (14.1), with movement away from the den to forage, (14.3), the model conservation equation d U d U
— + v ■ Ju = 0 ^ — = V- [Cu (x - Xu)u + Du (u)Vu]. (14.4)
We now have to consider appropriate initial and boundary conditions. Biologically realistic boundary conditions may involve local migration dynamics. However, the simplest possible boundary conditions are when we assume that wolves neither immigrate nor emigrate from the domain of interest denoted by Œ and which has to be determined. That is, we impose zero-flux boundary conditions for u, namely,
where n is the outward unit normal to the boundary, 9Œ, of the domain. Initial conditions, describing the expected spatial distributions of wolves at the beginning of a study period, is given by u(x, t) = u0(x). (14.6)
At any given time, the total number of wolves, Q, in the domain Œ is
Using (14.4) we see that
- i u(x, t ) d x = i d u(x, t ) d x =— i V- Ju dx = — i Ju ■ ni f Jo Jo Ot Jo JdO
So, the zero-flux boundary condition (14.5) guarantees a constant number of wolves in the pack within the domain Œ.
We obtain the average density, U0, of wolves in the pack throughout the region Œ
A Jo where A is the area of the territory O. The mathematical problem is now completely defined.
Suppose we consider the time-independent problem. Equation (14.4) becomes
By way of illustration let us consider the one-dimensional situation with the zero-flux boundary condition (14.5) and with the continuous convection form (14.2) and obtain the steady state density distribution and territory size as a function of pack size. Let us further take the density-dependent diffusion coefficient to be given by Du (u) = duun. The last equation then becomes, on integrating with respect to x, n du cuu tanh p(x — xu) + duu — = constant. (14.11)
Linear Diffusion, n = 0
Here D(u) = du a constant. Using zero-flux boundary conditions integration immediately gives the steady state solution, us (x), in one space dimension as
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