Math/ Comput. Modding, Vol. 12, No. I, pp. 105-112, 1989 Printed in Great Britain. All rights reserved

0895-7177/89 $3.00 +O.OO Copyright 0 1989 Pergamon Press plc

PERSISTENCE AND EXTINCTION IN MODELS OF TWO-HABITAT MIGRATION

H. I. FREEDMAN

Department of Mathematics, University of Alberta, Edmonton, Alberta T6G 2G1, Canada

(Received April 1986; accepted for publication May 1988)

Communicated by X. J. R. Avula

Abstract-It is the purpose of this paper to model a population living in two habitats with migration between the two habitats and predation during the migration process. In addition, harvesting could occur in one or both habitats. Models of two autonomous ordinary differential equations are utilized with predation incorporated as a probability of survival. A model of three equations is given which includes the predator dynamics. Persistence and extinction criteria are examined. An application to the Burwash caribou herd is given.

1. INTRODUCTION

Freedman and Waltman [l] proposed a model of a single population living in two habitats with migration between them across a barrier. The model was extended by Freedman et al. [2] to include the case where animals leaving one habitat did not necessarily reach the other habitat. In this case, the existence of a positive equilibrium as a function of barrier strengths was examined.

It is the main purpose of this paper to extend a special case (equal barrier strengths and survival probabilities) of the model considered in Ref. [2] to the case of harvesting in one and both habitats. A three-dimensional model will also be considered, modeling in addition to the above-mentioned populations the predator population. Two-dimensional predator-prey populations with dispersal have been considered by Chewning [3] and Holt [4]. In a single habitat, single-species models with harvesting have been analyzed by Brauer and Sanchez [5], whereas predator-prey models with harvesting have been discussed by Brauer and Soudack [&8].

We have in mind an application to the Burwash caribou herd, which lives on both sides of the Shakwak Trench in the Kluane mountains of the Yukon Territories, northern Canada. The dynamics of this herd have been discussed by Brown [9] and by Gautier et al. [lo]. The question of persistence or extinction of this herd is of importance to the economy of the region.

Persistence has been defined rigorously in Ref. [l l] (termed “strong persistence” there) and Ref. [12]. A population N(t) is said to be persistent if N(t,) > 0 aliminf,,, N(t) > 0. If, in addition, lim inf,,, N(t) B 6 > 0, for N(t,) > 0, N(t) is said to be uniform/y persistent. In Ref. [I 31, conditions were given under which persistence implies uniform persistence. If all populations of a system are (uniformly) persistent, then the system is said to be (uniformly) persistent.

In Refs [13, 141, the definition of (uniform) persistence was extended to sets with more general boundaries than the coordinate axes and/or planes. Such an extension is required for analysis of the models with harvesting and is tantamount to requiring that the omega limit set of orbits initiating in the interior of the set be a (uniform) positive distance from the boundary of the set.

2. TWO-DIMENSIONAL MODEL

We propose as a model of a population living in two habitats with migration and harvesting, the system of autonomous ordinary differential equations

105

106 H. I. FREEDMAN

x’ = x,g,(x,> - 4(x,) +@,(x,) - H,, x, >o max(pch,(x,) - H,, 0), x, = 0,

x’ = x,&(x,) - ch(x,) +M,(x,) - Hz, x2 > 0 max(pth,(x,) - H,, 0), x2 = 0, (1)

d x,(O)=x,,>O, x,(O)=x,,BO, ‘=-.

dt

We assume all functions are sufficiently smooth that existence, uniqueness and continuability of initial-value problems hold for all xlO, xZO B 0.

The interpretations and properties of the various components of this model are as follows (see Refs. [11, 121).

g,(x,) is the specific growth rate of xi, i = 1,2, in the absence of migration and harvesting. Specifically, we assume

g,(O) > O, gltxi) < O, 3K; > 0 s.t. g,(q) = 0. (2)

h,(x,) is the migration function, i.e. hi is proportional to the number of the population in the ith habitat that leaves per unit time. Hence,

h,(O) = 0, hi(x) > hj(0) > 0. (3)

The constants may be interpreted as follows: E > 0 is the inverse barrier strength and is deemed to be the same in both directions across the barrier; p is the probability that xi having left the ith habitat will arrive safely at the other habitat. If predation or emigration occurs during migration, then 0 <p < 1; H, is the constant rate harvesting for the ith habitat.

We now proceed to analyze the population dynamics in the case of no harvesting and the cases of harvesting in one and in both environments.

3. NO HARVESTING

In the case of no harvesting, model (1) becomes

x; =Xlgl(x,)-Eh,(X,)+P~~?(X*),

x; =x&(x,)-6(x*) +Pchr(x,). (4)

According to results obtained in Ref. [2], a positive equilibrium E*(xy(t), x$(c)) will exist for

0 G 6 < ~sl(OY~;(O) +&(OV;(O) + ](g,(O)hXO) -g*(0)h;(O))2

+ 4pZg,(0)g2(O)h;(O)h;(0)1}/2(1 -P2)~;@)~;(0) = E. (5)

Further, if E*(t) exists, it is globally asymptotically stable. Finally, lim,+E*(c) = (0,O).

4. HARVESTING OF x2 ONLY

We now suppose that harvesting occurs in the second habitat, but not in the first. Model (1) now assumes the form

x; =x&,(x,) - &(x1) +P&(x,)

-” = max(pth,(x,) - H,,O), i

x&x,) - ch,(x,) +P&(x,) - K, x* > 0 x2 = 0.

(6)

Let C, be the x, isocline and C, be the x2 isocline. C, is independent of the harvesting parameter, but does depend on L and is given by the equation

J&(x,) = &(x,) - xlgl(xl>. (7)

From the properties of g,, h, and h,, x2 is an increasing function of x, (see Ref. [2]).

A model of two-habitat migration 107

The isocline C, is obtained by solving

x,g,(x,) - Mx,) = Hz -P&(X,). (8)

From the properties of g,, h,, h,, C, will have two branches, one increasing and one decreasing.

A positive equilibrium will occur whenever C, and C, intersect. C, and C, may intersect twice,

once, or not at all (see Fig. 1) depending on L and Hz. (Note we do not consider critical cases where the isoclines are tangent.)

In the case where such an equilibrium exists (call it E*(x:, XT)) its stability is given by the eigenvalues of the variational matrix

V* =

[

x:g;(x:) + s,(x?) - ch;W) ch;(x:)

ch;(x:) x%(x:) +&(x3 -6&(x:) 1 If both c and H, are small, there is a unique equilibrium [see Fig. l(a)]. In this case E* is

asymptotically stable in the large. If 6 is small and H2 intermediate [Fig. l(b)] or t intermediate and H, small [Fig. l(d)], there are

two positive equilibria. One of them, denoted by E* is asymptotically stable in the large. The other is a saddle point whose stable manifold is denoted by r.

In the three cases where E* exists, persistence with respect to a set G occurs. All solutions initiating in the interior of G persist and in fact tend to E* as t +a. Figure 2 describes the set G in these cases. In all other cases, extinction of both populations occurs for all initial populations.

b) X2

Cl

Lk 1 C2

K2

0 5 %

K2 L C2

0 5 5

Fig. 1. Isoclines in the case of harvesting in the second habitat only. C, is the x, isocline, C, is the x2 isocline, r is the separatrix. Intersections of C, and C, are equilibria. (a) 6 small, H, small. (b) L small, H, intermediate. (c) L small, H, large. (d) E intermediate, H, small. (e) t intermediate, H2 large. (f) L large,

H,>O.

108 H. I. FREEDMAN

(a)

X2

(b)

1 0 Kl Xl

Fig. 2. The persistent set G. (a) 6 small, H, small. (b) t small, H, intermediate and L intermediate, H2 small

5. HARVESTING IN BOTH HABITATS

We now consider the full model (1). The isoclines (which intersect at equilibria) are obtained from the equations

and

&(x2) = H, +&(x1) - X,&(X,) (9)

xzgz(x,) - &(x,) = H, -p&(x,). (10)

Generically (again we do not consider critical cases) these isoclines, denoted C, and C,, respectively, may intersect in four, two or no points (see Fig. 3). In the case of intersection, one of these (denoted E*) will be asympotically stable in the large. In the case of two equilibria the other equilibrium will be a saddle point, whereas if there are four equilibria [Fig. 3(d-ii)], two of them are saddle points and the fourth is completely unstable.

As in the previous case, when the isoclines intersect in two or four points, there will be a separatrix (denoted r) such that solutions beginning “below” the separatrix intersect one of the coordinate axes. Solutions initiating “above” the separatrix tend to E* as r + cc. Hence, if G is again defined as the set of points in the first quadrant lying “above” or on the separatrix, system (1) exhibits persistence with respect to G.

If E* does not exist, extinction occurs.

6. BURWASH CARIBOU DYNAMICS

The two-dimensional model discussed in the previous sections simulates in a reasonable way many of the dynamics of the Burwash caribou herd [lo]. This herd occupies two habitats, the Brooks Arm Plateau located on the eastern side of the Shakwak Trench in the Kluane mountains of the Yukon Territories, northern Canada, and the Burwash Uplands on the western side (see Fig. 4).

The environment is closed in the sense that no caribou enter or leave the total habitat. However, the caribou do migrate across the Shakwak Trench from one habitat to another. Based on studies by Gautier et al. [lo], most crossings from mid-winter to calving are from the Brooks Arm Plateaux to the Burwash Uplands, whereas the reverse is true from the rut season (mid-September) to mid-winter.

A model of two-habitat migration

I (W , w

109

Fig 3. Isoclines in the case of harvesting in both habitats. C, is the x, isocline, r is the separatrix. Intersections of C, and C, correspond to equilibria. (a) L, H,, H2 small. (b) c, H, small, H2 intermediate. (c) c, H, small, H, large. (d) L small, H,, H2 intermediate; (i), (iii), two equilibria; (ii) four equilibria.

(e) t small, H, intermediate, H2 large. (f) c intermediate, H,, H2 small. (g) All other cases.

The Shakwak Trench is in a subalpine forest, and hence the caribou are subject to predation by wolves in the Trench. Hence there is a positive probability that an animal which leaves one of the habitats will not arrive safely at the other. However, since most animals cross safely, p > l/2, and further, since no animal is observed to cross the Trench more than twice a year, t can be thought of as “small”.

Native populations can (and do) harvest the caribou in both habitats. Non-natives can harvest caribou (under licence) in the Brooks Arm region only. However, a proposed change in law would allow licenced hunting in the Burwash Uplands. It has been conjectured by D. Gautier (personal

Fig. 4. Burwash caribou herd habitats.

110 H. I. FREEDMAN

communication) that since the herd is small (between 350 and 550 animals), and since it is already in a slow state of decline, any additional harvesting will cause the herd to become extinct.

From the analysis in the previous section even if the total population is in a slow decline, it may be tending towards a stable equilibrium. In that case a small additional amount of harvesting would not cause the herd to become extinct. If the herd population is “below” the separatrix, extinction will occur unless harvesting is reduced. If the population is “above” the separatrix, then sufficient additional harvesting will cause the separatrix to “move up” above the given population level, switching persistence to extinction.

7. THREE-DIMENSIONAL MODEL

At this time we present a mode1 of the prey population in two habitats, including the dynamics of a predator population. We assume that the predator lives only on the prey and that the predators do not enter or leave the environment. We also assume that no harvesting occurs. The mode1 to be considered is

x; = x,g,(x,) -&(x,) + 6(x,) -YP,(X,),

x; = X*&(X*) -&(x,) + &(x,) -YP,(X,),

Y’ = Y( -s + C,P,(X,) + C*P*(%)). (11)

This is a generalization of a mode1 considered in Ref. [l]. Here gi and hi are as before; pi(xi), i = 1, 2, are the predator functional response to the prey. We assume

P,(O) = 0, p:(xi)>O forxi>O, i=l,2. (12)

Note that predation is contained in separate terms, and hence p = 1 from previous models.

We are interested in the question of persistence. In order to apply the techniques utilized in Refs. [ 11, 121, we need to analyze all boundary equilibria. However, the only two such equilibria for t > 0 are &(O, 0,O) and E*(x y, XT, 0). Note that since p = 1, E* exists for all t > 0, and is globally stable in the x,-x2 plane.

The variational matrices at these equilibria are

I

&(O)-&(O) Eh;(O) 0

v. = CA;(O) g,(O) - th;(O) 0 1 (13)

0 0 --s

and

[

x:g;(x:) +8,(x:) - &(x:) ch;(x:) -P,(X?) v* = ch;(x?) xz*g;(xz*) + g,(x:) - th;(x:) -P2(X?)

0 0 -a + clPl(x:) + C*P2(Xz*) 1. (14)

Clearly from equation (13) E,, is stable in the y-direction. To show that E, is unstable in the x,-x2 plane, consider the characteristic polynomial

I”* - a,(c);l + a,(c), (15)

where

a,(c) = g,(O) + g,(O) - c@;(O) + h;(O)),

a2 (6) = gl (Ok, (0) - c (g, (OY; (0) + g2 (OV i (0)). (16)

A model of two-habitat migration III

Fig. 5. Dynamics near the coordinate planes of system (11). Solid arrows indicate directions in the planes. Dashed arrows indicate directions into the interior.

For

g1 (Ok, (0)

C iRi(0)MO) +g?(OV;(O)’ a,(c) > 0 and a*(c) > 0.

For

gl m3 (0) Sl(O> + g*(O) g,(WS(O) +gz(w;@) < c < h;(o) + h;(o) ’

U,(E) < 0 and ~~(6) -C 0.

For

!?I (Ok, (0)

h{(O) + hi(O) < c, a,(c) < 0 and az(c) < 0.

Hence in all cases V, has at least one root with a positive real part. E* is globally asymptotically stable with respect to the interior of the positive x,-x1 plane. We

will require E* to be unstable in the y-direction. From equation (14) this will be the case provided

--s + c,p,(xT) + C*Pz(-G) > 0. (17)

By techniques similar to those used in Refs. [l 1, 121 and utilizing results from Ref. [13] we have the following theorem:

Theorem 1. Let inequality (17) hold. Then system (11) is uniformly persistent.

The dynamics near the coordinate planes are shown in Fig. 5.

8. DISCUSSION

A model of a population living in two habitats is proposed, for the case where the population migrates between the habitats, and that predation occurs during the migration. This model is applied to the dynamics of the Burwash caribou herd. Analysis of the model indicates that if the herd persists under the present harvesting laws, a sufficiently small increase in harvesting will not cause extinction. However, if the harvesting is too large, the herd will become extinct.

In the case that the predator lives solely on the prej (which does not happen in the Burwash dynamics) a three-dimensional model is proposed and analyzed. The case of no harvesting is analyzed, and a criterion for persistence is given. The case where there is harvesting is considerably more complicated and will be considered in a later paper.

Acknon,/edgemenrs-This research was partially supported by the Natural Science and Engineering Research Council of Canada, the Central Research Fund of the University of Alberta and the Boreal Institute of Northern Studies.

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112 H. I. FREEDMAN

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