A Single Species in One Spatial Dimension

Transcription

1 Lectre 6 A Single Species in One Spatial Dimension Reading: Material similar to that in this section of the corse appears in Sections 1. and 13.5 of James D. Mrray (), Mathematical Biology I: An introction, 3rd edition, Springer Interdisciplinary Applied Mathematics Series, Vol. 17, ISBN Available online at with frther discssion in Chapter of James D. Mrray (3), Mathematical Biology II: Spatial Models and Biomedical Applications, 3rd edition, Springer Interdisciplinary Applied Mathematics Series, Vol. 18, ISBN Sprce bdworms The methods we ve developed to stdy two interacting species can also be adapted to the stdy of a single species whose poplation varies in both space and time. Models of this sort are sometimes called 1+1 dimensional as they involve derivatives in one space and one time dimension. Or primary example will be a model for the growth of sprce bdworms, a family of insect species whose larvae ravage the forests of North America. Tree-ring stdies 1 sggest that serios otbreaks have been happening every few decades since at least the 169 s. We ll stdy two models for these insects: first, an ODE that describes the dynamcis of a small, isolated poplation of bdworms and then, second, a PDE model whose analysis sing phase-plane methods similar to those the ones we ve developed for two-species models will enable s to discss strategies for containing the pest. 1 Swetnam and Lynch (1993), Mlticentry, Regional-Scale Patterns of Western Sprce Bdworm Otbreaks, Ecological Monographs, 63(4):

2 p(n) B B/ A A 3A 4A N Figre 6.1: Predation p(n) as a fnction of bdworm poplation N. Note that when N A the crve is qadratic, so that near zero it is almost flat The ODE model The basic model is dn dt = rn 1 N K BN A + N (6.1) where the parameters r, K, A and B are all positive real nmbers. The first term is a standard logistic growth law, while the second describes predation. It s instrctive to give the predation term a name, say, p(n), and to rewrite it in sch a way as to bring ot the role of the parameter A, which has nits of bdworms : p(n) = BN A + N = (1/A) (1/A) BN A + N = B(N/A) 1+(N/A). This form makes it clear that when N = A, the predation term is P (A) =B/, and so B has nits of (bdworms/time). Now consider the limits of small and large N. When N A, p(n) isapproximatelyaqadratic, B(N/A) 1+(N/A) B N, A bt when N A we have lim N!1 B(N/A) 1+(N/A) = B. Ptting all these observations into a sketch yields Fig Ths when there are jst a few bdworms, the predation term says that the predators don t bother eating them, bt when there are a lot of bdworms, the predation term levels o to a constant vale. The ecological idea behind this behavior is that the pool of predators birds, mainly is finite and their appetites are bonded: even when presented with a forest fll of food, the birds can only eat some fixed, thogh perhaps very large, nmber of bdworm larvae per day. 6.

3 Dimensionless form If one defines dimensionless variables = N A and = B t A then the sal sorts of calclations rece Eqn. (6.1) to d = 1 q 1+. (6.) where the new dimensionless parameters are = ra B and q = K A. Here q is a dimensionless version of the carrying capacity. Eqilibria The eqilibrim condition for (6.) is /d =or,eqivalently,?? 1 =?. q 1+? As we re mainly interested in eqilibria? >, we can divide both sides of this expression by? to obtain F () 1 = q 1+ G() where I have dropped the? s and defined two new fnctions: F () andg(). One can gain insight into the pattern of eqilibria by sketching F () andg() onthe same axes: Fig. 6. shows several examples. Note that F () isespeciallyeasyto sketch as it s jst a line passing throgh the points (, ) and(q, ), while G() has G() =, attains a local maximm at =1and,for 1, satisfies G() 1/. We will focs on cases where there are three nonzero eqilibria that I ll label < 1 < < 3. The ODE can be written d = (F () G()) and so the sign of /d, which determines the stability of the eqilibria, is the same as that of (F () G()). As Figre 6.3 illstrates, the stable eqilibria are 1 and 3, which are referred to as the refge and otbreak eqilibria, respectively. The ecological reasoning behind these names will become clearer toward the end of the lectre. 6.3

4 (q a, ) 1 3 F(), G() F(), G() F(), G() F(), G() 1 = a 1 = a Figre 6.: Plots showing the crves F () and G() for choices of and q that lead to one (pper left), two (bottom row) or three (pper right) eqilibria for the ODE (6.1). Note that the lengths of the horizontal and vertical axes vary between panels. Bdworm Eqilibria ρ F() G() 1 3 q refge otbreak Figre 6.3: The crves F () and G() for =.55, q =7and a diagram indicating the positions of and stability of the three nonzero eqilibria < 1 < < 3. The stable ones are 1, the refge eqilibrim, and 3, the otbreak eqilibrim. 6.4

5 6.1. Adding spatial strctre So far we have modelled the bdworm poplation as a fnction of time alone, bt now we will introce spatial variation, replacing ( ) with(x, ) withx (,L). Here the fnction (x, ) representsapoplationdensity bdwormspersqare meter, say, or perhaps bdworms per sprce tree. Real forests aren t one-dimensional, bt we might imagine (x, ) todescribea large forest that had been sprayed in sch a way that long strips of it had become poisonos to bdworms. The bdworms cold still live in gaps between the sprayed strips, bt then their poplation wold vary more strongly across the gap than along it. In this case we d take L to be the width of the gap and impose the bondary conditions (, ) ==(L, ) toindicatethattheforestotsidethegapistoxic to bdworms. If we assme the bdworms di se in space, as well as reprocing locally as described by the ODE model, we end p with the following = D@ xx {z } di sion + 1 q {z 1+ } local dynamics. (6.3) where D, the di sion coe cient, is a positive constant. 6. A mathematical interlde Eqn. (6.3) is a special case of a whole class of models of the = D@ xx + f(). (6.4) The analog of an eqilibrim for this sort of system is a steady state soltion to the PDE: one for =. This means that (x, ) isactallyindependentof time, so (x, ) =(x) andthepdeineqn.(6.4)becomesanodeinx: =D d + f(). (6.5) dx Typically one can only solve this sort of eqation nmerically, bt one can get some insight into the form of the soltions by first converting (6.5) into a system of two first-order ODEs by defining v = /dx and sing an approach from one of the Problem Sets: dx = v ˆf(, v) and dv dx = d dx = 1 f() ĝ(, v), (6.6) D where I have defined the fnctions ˆf(, v) andĝ(, v) toemphasisethatthissystem derived from the steady-state condition (6.5) looks very mch like a two-species model and is certainly amenable to the sort of phase plane analysis we ve been doing in recent lectres. Eqilibria of (6.6) satisfy dx = v = and dv dx = 1 f() =. D 6.5

6 and so look like (?, ) where f(? )=. The linear stability analysis of sch eqilibria is especially straightforward. The relevant matrix of partial derivatives is J = v ĝ = apple 1 (1/D)f (? ) (6.7) which has =Tr(J) = and =det(j) = f (? ) D and so has either two real eigenvales of opposite sign when f (? ) < or two pre imaginary eigenvales when f (? ) >. This means that eqilibria of (6.6) at which f (? ) < aresaddles,whilethoseatwhichf (? ) > arecentres. Althogh the presence of a centre sally means that linear stability analysis is inconclsive, systems sch as (6.6) have a formal resemblance to the eqations of classical mechanics and we can exploit this observation to constrct a constant of the motion of the form H(, v) = 1 Dv + () where () = Z f(s) ds. (6.8) This fnction is constant along crves traced ot by soltions ((x),v(x)) to (6.6), as one can readily verify with the following comptation: H ((x), v(x)) = dx + f() = vf()+ D = vf() f()v =. (Dv) Ths soltions ((x),v(x)) trace over contors of the fnction H(, v): we ll conclde the lectre by applying this idea to the spatially-extended bdworm model. 6.3 Steady-state soltions for the bdworm PDE If we take the local dynamics from the bdworm system, Eqn. (6.5) becomes =D d dx + 1. q 1+ Applying the approach from the previos section leads to the system dx = v and dv dx = 1 apple 1, (6.9) D 1+ q which can have two, three or for eqilibria, depending on the vales of the parameters and q. In the case where there are for eqilibria they will be (, ), ( 1, ), (, ) and ( 3, ), 6.6

7 v L v-.4 (a) Contors of the fnction H(, v). The dashed crves are the contors that pass throgh the eqilibria of the system in Eqn (6.6) v.3 (b) Vale of the length integral in Eqn (6.11) as a fnction of v Figre 6.4: Here the parameters are D =1, =.55 and q =7, which means that the corresponding ODE model in Eqn (6.) has three nonzero eqilibria, 1 < < 3. The singlarities in the length fnction occr for contors that pass throgh the eqilibria ( 1, ) and ( 3, ). where < 1 < < 3 are the eqilibrim poplations for the simple, spaceindependent ODE model discssed in Section and illstrated in Figre 6.3. The constant of the motion associated with steady states of the bdworm PDE is H(, v) = 1 Dv + q 3 +arctan() (6.1) Contors, soltions and lengths Sppose that we re planning to spray the forest in strips, leaving gaps of width L: then we re interested in steady-state soltions to Eqn (6.3) with the bondary conditions () = (L) =. Oneexpectsthesesoltionstobesymmetricand bmp-shaped, with a single maximm at x = L/ (see Figre 6.5b), and so they ll also satisfy > and <. dx x= dx x=l In terms of the system of ODEs in Eqn (6.9), this means that we want to look for soltions with initial conditions () = and v() = v >. These conditions correspond to points on the positive v-axis in the contor map that appears at left in Figre 6.4 above. And as we vary x across the interval apple x apple L, thepoint((x),v(x)) will trace over a contor of H(, v), eventally crossing the -axis and stopping on the negative v-axis. Unfortnately this bsiness of tracing over contors means that the role of the space variable x is only implicit 6.7

8 .4 v (a) Contors corresponding to the steady state soltions at right. The dashed crves are contors that pass throgh the saddles of the system in Eqn (6.9) x (b) Bmp-shaped steady-state soltions to the PDE in Eqn (6.3). The intervals over which the soltions are plotted have been shifted so they re centred on zero, while the dashed horizontal lines pass throgh the eqilibrim vales 1 < < 3 of the nderlying ODE model (6.). Figre 6.5: The solid, colored crves at left correspond to the steady-state soltions at right. The parameters are D =1, =.55 and q =7, which means that the corresponding ODE model given by Eqn (6.) has three eqilibria: 1 < < 3. in the contor map and so it s not immediately clear which contor (or contors, as there may be more than one) correspond to a steady-state soltion for an interval of a given length. Instead, we are obliged to fix initial data () =, v() = v > andthen find the corresponding soltion ((x),v(x)) and compte nmerically the vale L sch that (L) =or,eqivalently,findthevaleofx at which the crve traced over by ((x),v(x)) hits the negative v-axis in a contor plot sch as Figre 6.4. More generally, one can ask when (that is, for which vale of x) thesoltioncrve ((x),v(x)) reaches an arbitrary point ( 1,v 1 )onthecontorofh(, v) thatpasses throgh ((),v()) = (,v ). The answer is given by an integral x = Z v1 v dv dv/dx along the contor that connects (,v )to( 1,v 1 ). In particlar, the length L of the interval whose steady-state soltion has ((),v()) = (,v ) is the length of the arc rnning from (,v )to(, v )andisgivenby Z L = v v dv dv/dx. (6.11) For the bdworm PDE, with H(, v) givenbyeqn6.1,it snotpossibleto do this integral by hand, bt one can evalate it nmerically: Figre 6.4b shows 6.8

9 typical reslts. It s a graph showing the length of the interval associated with the initial data ((),v()) = (,v )asafnctionofv. Notice that the length integral diverges for two vales of v : these are the vales that correspond to the contors that pass throgh the points ( j, ), which are eqilibria of the ODE system (6.6) Control of the pest It s possible to interpret the steady-state soltions of the PDE in terms of the soltions to the dimensionless ODE model given by Eqn 6.. For s ciently small domains with the parameters sed to draw Figres 6.4 and 6.5 s ciently small means L less than abot 4 there is only one soltion for a domain of size L. Frther, the corresponding contor lies entirely to the left of the point ( 1, ) and hence the bdworm poplation remains below 1 throghot the region. Ecologists refer to this as a refge soltion becase althogh there are still bdworms present (the strip provides them with a refge from the spraying), their poplation doesn t explode to the point that it threatens the forest. The smaller of the two bmps in Figre 6.5b is an example. If we keep the gaps between or sprayed strips narrow enogh that only refge soltions are possible, the forest shold be safe. By contrast, if L is too large there are three sorts of steady-state soltion: one whose contors begin jst above the one that passes throgh ( 1, ), a second one that begins jst below this contor and a third whose contor lies close to the one passing throgh ( 3, ). The first of these is nstable in the sense that small pertrbations (proced by, say, the arrival of wind-blown bdworms) will case a poplation boom and the bdworm poplation near the middle of the strip will bild p almost to the level of the eqilibrim 3. These sorts of steady-states are called otbreak soltions becase they involve very large bdworm poplations: the larger of the two bmps in Figre 6.5b is an example. 6.9