Multi Wolf Pack Territorial Model

Response to neighboring packs occurs primarily through RLU marking. The particular nature of the response (in terms of wolf movement) is not well understood and has been investigated in two different ways. In the first, the presence of foreign RLU marks increases the speed of movement back towards the den (central territory) area while increasing the production of familiar RLUs. In the second, wolves respond to gradients in foreign RLU markings by moving away from regions of high density at the same time increasing production of their own scent marks. Although similar behaviours are observed with both these scenarios there are some differences which we come back to below.

Because RLUs are made by a few mature dominant wolves in each pack, the location of these wolves is key in determining the RLU marking patterns. For the purposes of this model we can describe the location of such a dominant wolf by a probability density function denoting the chance of finding the wolf at point x and time t. For any given pack, we sum these probability density functions over the number of RLU-marking wolves. This provides a measure of the expected density of RLU-marking wolves in the pack at a point x and time t. From now on we refer to this quantity as the expected local density of wolves in a pack.

For a model involving two adjacent, interacting (in effect competing) wolf packs, the relevant state variables are the expected local densities of wolves in pack number 1, u(x, t); wolves in pack number 2, v(x, t); RLUs from pack number 1, p(x, t); and RLUs from pack number 2, q (x, t).

We must now include equations for the RLU densities which reflect the wolf responses to foreign RLUs from other packs. Based on the above, we assume that when members of a pack encounter RLUs from an adjacent pack, they move away from these foreign RLUs and back towards the den while also increasing their rate of RLU marking. Although mortal strife may occur when adjacent packs interact, for the purpose of modelling the populations we assume that such fatal interactions are very rare and that the number of wolves remains constant over the time period of the model. Remember that we are only considering the summer months.

The word equation for the wolf dynamics (of pack 1) is now

Rate of change in expected density of wolves (pack 1)

= Rate of change due to movement of pack 1 wolves towards their den

+ Rate of change due to dispersal of pack 1 wolves

+ Rate of change due to movement of pack 1 wolves away from the RLUs made by pack 2

the terms of which we must now quantify. Let us first consider movement in response to foreign RLU markings. We consider two ways to model movement induced by RLU

levels. In the first, the response is assumed to increase the rate of movement back towards the den site. At the most extreme, this movement is assumed only to occur in the presence of competing RLUs but can be modified to allow movement independent of neighboring packs. In either of these, the convection flux, JCu, described in the last section, (14.1) is modified to

where, to show the dependence on foreign RLUs, q, we write as cu (x — xu, q) which is a function of q such that dcu /dq > 0 since in the presence of foreign RLUs, the wolves retreat towards the den site. The function cu (x — xu, q) is typically a bounded monotonically increasing function of q; a function qualitatively like Aq/(B + q) with A and B constants is reasonable.

In the second case, the response to RLUs is to make the wolves move down gradients of foreign RLU density. In this case, the movement is modelled mathematically by a flux, Jau given by

where au (q) is another monotonically non-decreasing function. Gathering these together we now have the conservation equation for the wolves in pack 1 as du + V • J + Jdu + Jau ] = 0, (14.18)

where the fluxes are given by

Jcu = —ucu (x — xu Jdu = —du (u)V u, Jau = au (q )uV q,

The equation for movement of the second wolf pack mirrors that for wolf pack 1 and is given by

Rate of change in expected density of wolves (pack 2)

= Rate of change due to movement of pack 2 wolves towards their den

+ Rate of change due to dispersal of pack 2 wolves

+ Rate of change due to movement of pack 2 wolves away from the RLUs made by pack 1

and is represented mathematically as

, q), cu (0) > o, du (0) > 0, au (0) > 0, dcu dq ddu du dau dq

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