J. Math. Biol. (27) 55:575 64 DOI 1.17/s285-7-96-4 Mathematical Biology Dispesal and settling of tanslocated populations: a geneal study and a New Zealand amphibian case study Abbey J. Tewenack Key A. Landman Ben D. Bell Received: 5 Decembe 26 / Revised: 11 Mach 27 / Published online: 11 May 27 Spinge-Velag 27 Abstact Tanslocations ae widely used to eintoduce theatened species to aeas whee they have disappeaed. A continuum multi-species model famewok descibing dispesal and settling of tanslocated animals is developed. A vaiety of diffeent dispesal and settling mechanisms, which may depend on local population density and/o a pheomone poduced by the population, ae consideed. Steady state solutions ae obtained using numeical techniques fo each combination of dispesal and settling mechanism and fo both single and double tanslocations at the same location. Each combination esults in a diffeent steady state population distibution and the distinguishing featues ae identified. In addition, fo the case of a single tanslocation, a elationship between the adius of the settled egion and the population size is detemined, in some cases analytically. Finally, the model is applied to a case study of a double tanslocation of the Maud Island fog, Leiopelma pakeka. The models suggest that settling occus at a constant ate, with epulsion evidently playing a significant ole. Mathematical modelling of tanslocations is useful in suggesting design and monitoing stategies fo futue tanslocations, and as an aid in undestanding obseved behaviou. A. J. Tewenack K. A. Landman (B) Depatment of Mathematics and Statistics, Univesity of Melboune, Melboune, VIC 31, Austalia e-mail: k.landman@ms.unimelb.edu.au A. J. Tewenack e-mail: a.tewenack@ms.unimelb.edu.au B. D. Bell Cente fo Biodivesity and Restoation Ecology, Victoia Univesity of Wellington, P. O. Box 6, Wellington, New Zealand e-mail: Ben.Bell@vuw.ac.nz
576 A. J. Tewenack et al. Keywods Dispesal Settling Population Multi-species continuum model Tanslocation Consevation Leiopelma pakeka Mathematics Subject Classification (2) 35K55 35K57 35K65 1 Intoduction Tanslocations ae widely used to eintoduce theatened species to aeas whee they have since disappeaed, and to move them to aeas whee theats can be moe easily contolled [27]. Many tanslocation attempts have failed [18] often due to insufficient planning o feasibility studies. When dealing with theatened species, Bidgewate et al. [8] comment that an ad hoc appoach is not ethical. Consequently thee is a need fo data fom successful tanslocations to be used to maximum effect. Data collected fom caefully designed tanslocations povides vital insight into the behaviou of the species and its pedicament [1]. Coupled with mathematical models, this data becomes a valuable planning tool to ensue the success of futue tanslocations. Mathematical modelling of tanslocations is useful in two majo ways; fistly to suggest design o monitoing stategies fo futue tanslocations, and secondly as an aid in undestanding obseved behaviou. Rout et al. [26] and Tenhumbeg et al. [31] discuss methods fo detemining optimal stategies fo tansfeing animals fom a captive population to one o moe wild populations. Unlike these models that conside the numbe of animals in each population, ou inteest is in the spatial distibution of tanslocated populations; fo example, what size aea will be equied to tanslocate 1 animals? In planning a tanslocation, it is impotant to conside the mode of dispesal of the species in ode to ensue that the intoduced population does not spead futhe than anticipated [14]. Using a spatial model in paallel with decision-making tools such as those descibed in [26] and [31] will povide plannes of futue tanslocations with a solid foundation on which to base decisions. Mathematical modelling of tanslocations can also be of use in helping zoologists undestand animal behaviou. In paticula, double tanslocations (such as the case study we discuss in Sect. 8) povide useful infomation about the inteaction between two populations of the same species, especially if these populations inteact in a way that is not intuitively clea. Models of inteacting intaspecific populations appea fequently in the liteatue [6,7,23,29,36]. Howeve, seveal diffeent models fo dispesal aise in the liteatue [13], each pedicting that two populations eleased at the same position at diffeent times will inteact in diffeent ways. We wish to utilise these diffeences by compaing modelling esults with data fom a double tanslocation to deduce the most likely dispesal mechanism fo the species being consideed. Tanslocations involving a elease of a numbe of animals at a single location ae investigated hee. We ae inteested in the way animals dispese and choose a site in which to settle down. While we have chosen to use continuum models fo this pupose, othe appoaches such as integodiffeence equations have been successful in modelling simila poblems such as invasions and pedato-pey dynamics [16,21]. The pocess of dispesal coupled with settling has been studied in othe contexts.
Dispesal and settling of tanslocated populations 577 Boadbent et al. [9] and Williams [35] conside a specific poblem in the context of the distibution of insect lavae. Shigesada [28] pesents an analytic solution to a elated poblem whee the dispesing animals ae also unde the effect of population pessue (see Sect. 6.2). A simila poblem, studied by Edelstein-Keshet [1] and Mois et al. [19], aises in the context of hebivoous insects seaching fo suitable food plants. We would like to genealise these esults by compaing diffeent dispesal mechanisms and diffeent settling mechanisms, and obseving thei effects on the esulting animal distibution. Ou motivation fo consideing this poblem is twofold; (i) to undestand dispesal and settling mechanisms in geneal and (ii) to suggest tanslocation stategies fo the theatened Maud Island fog Leiopelma pakeka, a ae New Zealand species that until ecently inhabited a single island in the Malboough Sounds [5]. To ensue the suvival of this fog, and othe endemic species, a Native Fog (Leiopelma spp.) Recovey Plan was developed by the New Zealand Depatment of Consevation in 1996 [22,32]. Befoe caying out any lage-scale tanslocations, tial tanslocations wee caied out within Maud Island. In 1984 and 1985, 1 fogs wee tansfeed to a new location on the island in two goups, eleased a yea apat [4]. All fogs wee eleased at the same point and the population has been obseved ove a 22-yea peiod. Figue 1 illustates the data collected by Bell et al. [4]. The aveage distance the fogs tavelled befoe settling was significantly lage fo the second eleased population than fo the fist population. This indicates that the second population was awae of the pesence of the fist population. But how was this infomation communicated? The fogs ae known to engage in only vey limited vocal communication and lack extenal eadums (tympana). Howeve, Lee et al. [17] and Waldman et al. [33] showed that these fogs can communicate via faecal chemosignals. We ae theefoe pompted to ask the following questions: 1. Is the spatial sepaation due to some fom of communication between fogs, o athe that the envionment can only suppot a maximum density of animals? 2. If communication plays a ole in dispesal and settlement, in what fom o foms? 3. If thee is some inteaction, does it affect the dispesal behaviou of the fogs, the pocess of selecting a suitable place to settle, o both? 8 Numbe of fogs 6 4 2 5 1 15 2 25 Distance (m) Fig. 1 The distance fom the point of elease to the cente of activity of the new home ange in tanslocated fogs. Black: fist elease; ed: second elease. With data povided by B. Bell (modified vesion of a Figue in [4])
578 A. J. Tewenack et al. Motivated by these questions, we pesent a suite of continuum models fo tanslocated animal populations epesenting a ange of undelying dispesal and settling behavious. Numeical solutions ae given, while analytic solutions ae povided fo some cases. We discuss qualitative featues of the esulting distibutions that can be used to identify the most appopiate model fo a paticula data set. Finally, we conside the above case study: a double tanslocation of the Maud Island fog. We compae data with modelling esults and discuss the implications to ou undestanding of the behaviou of the fog population. We also highlight ways in which esults can be used to design futue tanslocations. We begin by intoducing the geneal fom of the model, and then follow this by specifying the diffeent dispesal and settling mechanisms. 2 Models A continuum multi-species appoach is taken in ode to diffeentiate between dispesing and settled animals. Motivated by ou inteest in the two pocesses, dispesal and settling, we divide the animal population into two subpopulations: those that ae in the pocess of dispesing, and those that have settled. A consevation equation is witten fo each of these two subpopulations. The Maud Island fog is a K-selected species, which means that the species is long lived and has delayed matuity and epoduction [4]. We conside animals that occupy discete home anges that ae small elative to the distances ove which the animals dispese, as is the case fo the Maud Island fog [4]. Theefoe thee majo assumptions ae made: 1. Thee ae no biths o deaths duing the dispesal and settling peiod, so the total numbe of animals is conseved ove time. 2. Tansfes fom the dispesing subpopulation into the settled subpopulation ae possible. 3. Tansfes fom the settled subpopulation into the dispesing subpopulation ae not possible; that is, settled animals ae assumed to emain stationay. As well as accounting fo dispesing and settled subpopulations, we ae inteested in modelling a double tanslocation. We next set up the consevation equations fo the fist and second tanslocations. 2.1 The fist tanslocation We define u to be the total animal population density of the fist tanslocation. We divide this up into two subpopulations. We denote the population density of dispesing (o motile) animals by u m, and the population density of settled o stationay animals by u s. Theefoe u = u m + u s. Assuming the animals ae moving in a two-dimensional egion which is adially symmetic about the point of elease, these densities ae functions of the distance fom the point of elease and time t.
Dispesal and settling of tanslocated populations 579 The system of consevation of mass equations has the fom u m + 1 (J u) = S u, (1) u s = S u, (2) whee the flux tem J u and the settling tem S u may be functions of the total population density u and pheomone concentation c u poduced by this population. A pheomone is a chemical compound poduced and seceted by an animal, that influences the behaviou and development of othe membes of the same species.the foms of J u and S u will be discussed in Sects. 3 and 4. The equations used to model the pheomone concentation c u ae given in Sect. 3.2.1. This system is solved with no-flux bounday conditions at. The numbe of individuals initially eleased is Q u, giving an initial condition appopiate fo the system (1) (2) as δ() u m (, ) = Q u δ() = Q u 2π, u s(, ) =. (3) Steady state conditions ae u m () = and u s = ū s (). Figue 2a and b show the initial elease at position = and a schematic of the steady state pofile espectively. Of couse the steady state evolves in the limit as t ; howeve, afte a cetain peiod of time, the solution appoaches the steady state within a specified toleance. We assume that the second tanslocation occus afte the fist elease has settled. 2.2 The second tanslocation The total population density of the second tanslocation of animals is denoted by v = v m + v s, whee v m and v s ae the motile and settled subpopulations espectively. Density u Density u () s Density v u s() Density u s() v () s t = Distance Distance Distance (a) (b) (c) Distance (d) Time Fist tanslocation Second tanslocation Fig. 2 Schematic epesentation of a double tanslocation. a Fist population, u m, is eleased at = at t = (u s (, ) = ). The black aow epesents the function Q u δ(). b Population u evolves accoding to the model and eventually eaches steady state distibution with ū s () and u m =. c Second population, v, is eleased at = at some late time, afte all of the fist population has settled to ū s ().Theed aow epesents the function Q v δ(). d Population v evolves accoding to the model (while u emains stationay) and eventually eaches a steady state distibution with v s () and v m =
58 A. J. Tewenack et al. The consevation equations have the same fom as those fo the fist tanslocation, namely v m + 1 (J v) = S v, (4) v s = S v, (5) whee the flux tem J v and the settling tem S v can depend on the fist animal population and its pheomone, as well as on the second population. In paticula, the inteactions with the existing settled fist tanslocated population u s =ū s () ae built into this system. Ou concen is with the natue of this inteaction and how it pemits diffeent elationships between the ū s () and v s (). We conside diffeent foms of J v and S v, which detemine how populations u and v inteact. The numbe of individuals initially eleased is Q v, so that the initial condition appopiate fo the system (4) (5) is v m (, ) = Q v δ() = Q v δ() 2π, v s(, ) =, (6) with no-flux bounday conditions at. The steady state has v m () = and v s = v s (). Figue 2c and d show the initial elease at position = and a schematic of the steady state pofile espectively fo the second elease. 3 Dispesal mechanisms In ode to examine the effects of diffeent dispesal mechanisms, we conside each mechanism in isolation. Of couse, in any application, dispesal may be the esult of two o moe mechanisms acting simultaneously, but we wish to identify featues paticula to each. Mechanisms which ae independent o dependent on a pheomone concentation ae both consideed. We wite the geneal fom of the flux of the fist and second tanslocations due to dispesal as u m J u = DD u, J v m v = DD v χv c u m. (7) The fist tem is a diffusive flux mechanism, whee D is the diffusivity of the dispesing animals and D u and D v ae dimensionless functions as yet unspecified. These tems coespond to andom movement of the animals. The second tem in J v is a epulsive tem due to the unfamiliaity of the second tanslocated population with the pheomone poduced by the fist tanslocated population. This is natually descibed as a chemotactically diected tem, which acts as a epulsive tem (when χ>). We fist discuss diffeent foms of D u and D v.
Dispesal and settling of tanslocated populations 581 3.1 Dispesal mechanisms independent of pheomone We fist discuss two diffusive flux mechanisms which ae not affected by the pesence of a pheomone. 3.1.1 Dispesal mechanism A: linea diffusion We conside a diffusive flux mechanism whee D u = D v = 1[11,2,24,3]. This model coesponds to a system whee dispesal occus as a esult of each individual taking an unbiased andom walk. Thee is no effect fom othe animals. 3.1.2 Dispesal mechanism B: population-dependent diffusivity An inceasing degeneate nonlinea diffusion is the most commonly adopted fom, whee the diffusivity is a function of the total population pesent. Hence, fo the fist and second tanslocated populations ( ) u n ( ) u + v n D u =, D v =, (8) u whee n > and u is an appopiately chosen scaling facto. By way of example, we conside n = 1. This is simila to the biased andom motion model of Guney and Nisbet [12] and the model studied by Shigesada [28]. Howeve, fo ou system the diffusivity is a function of the total population u and not just the motile subpopulation u m as in Shigesada [28]. u 3.2 Dispesal mechanisms dependent on a pheomone Reseach by Lee et al. [17] and Waldman et al. [33] indicated that Leiopelma pakeka communicates via faecal chemosignals. Thei wok suggests that the fogs can diffeentiate between the faeces of familia and unfamilia fogs, and Lee et al. suggested that fogs intuding into unfamilia teitoy would be epelled. Motivated by this idea, we conside two dispesal mechanisms that depend on a pheomone concentation. We begin by discussing the model fo the pheomone concentation itself. 3.2.1 Pheomone Lee et al. [17] obseved that the fogs in the expeiments defecated thoughout the day, and indicated that in the wild this would esult in fogs making thei home anges. They also found that the esponse of fogs to the pheomone deceased ove time. They suggested that this could be due to the pheomone decaying ove time, the fogs becoming accustomed to the pheomone, o the fogs speading the pheomone as they moved aound.
582 A. J. Tewenack et al. Guided by these obsevations, we assume that the pheomone is poduced at a constant ate by all animals. We also assume that it diffuses with constant diffusivity, and degades linealy ove time. We denote the pheomone poduced by animals in the fist and second tanslocated populations by c u (, t) and c v (, t) espectively. The equations descibing the pheomone concentations ae c u c v 1 = D c 1 = D c ( c ) u + ku λc u, (9) ( c ) v + kv λc v, (1) whee D c is the diffusivity of the pheomone, k > is the ate at which the pheomone is poduced, λ> is the decay ate of the pheomone, and u and v ae the total density of the fist and second tanslocation populations espectively (u = u m + u s and v = v m + v s ). In the following two sections we conside two ways in which the pheomone concentation could affect the dispesal pocess. 3.2.2 Dispesal mechanism C: nonlinea diffusion with pheomone-dependent diffusivity (chemokinesis) We conside the case whee the diffusivity inceases with the powe of the concentation of pheomone pesent. Hence, we wite D u = ( cu c ) n ( ) cu + c n v, D v =, (11) whee n > and c is an appopiately chosen scaling facto. By way of example, we conside n = 1. The D u only depends on c u because dispesal of the fist population ceases befoe the second population is eleased. Howeve, the second population will be influenced by pheomones poduced by the settled steady state of the fist population of unfamilia animals ū s () and thei own familia population. By choosing these foms fo the dimensionless diffusivity functions we ae consideing a case whee the pesence of pheomone impats some infomation about the local population density to neaby animals. The natue of the infomation depends on the choice of paametes in the patial diffeential equation fo the pheomone density. The choice of paametes is discussed in Sect. 7.3. 3.2.3 Dispesal mechanism D: epulsion (negative chemotaxis) While dispesal mechanism C consides a situation whee the pheomone inceases motility, hee we look at a case whee the pheomone has a epulsive effect. The motivation fo this choice of flux mechanism comes fom the peviously discussed expeiment conducted by Lee et al. [17] whee fogs geneally avoid a egion maked by an unfamilia fog. c
Dispesal and settling of tanslocated populations 583 Table 1 Summay of dimensionless diffusivities D u and D v fo the diffeent dispesal mechanisms Label Mechanism D u D v A Linea diffusion 1 1 u B Population-dependent diffusivity u+v u u c u c u + c v C Pheomone-dependent diffusivity c c D Repulsion (with χ>in(7)) 1 1 To model this, some assumptions must be made about the familiaity of animals in the two populations. Nothing is known about the elationships between animals when they wee collected, but they came fom acoss a wide aea of the souce population, athe than all fom a paticula site within that population (B.D.Bell, pesonal obsevations). We assume that the animals in the fist tanslocation ae unfamilia to those in the second. Theefoe it follows that the epulsive mechanism is the dominant dispesal mechanism fo the second population and not the fist. Indeed, the fist population is tanslocated into an aea fee of thei own species, and theefoe thee is no unfamilia pheomone pesent. Hence, the dispesal of the fist population is due to andom movement only, giving J u = D u m, J v = D v m χv c u m, (12) The epulsion model is based on Kelle and Segel s chemotaxis model [15]. The second tanslocated population dispeses by linea diffusion (dispesal mechanism A) and epulsive chemotaxis, so that χ>, while the fist tanslocated population dispeses by linea diffusion alone. Since dispesal mechanism A is investigated sepaately, we ae inteested in cases whee behaviou is dominated by the epulsive flux mechanism athe than diffusion, and choose paamete values accodingly. Table 1 gives a summay of the dispesal mechanisms discussed. We will efe to the dispesal mechanisms pimaily by thei labels A D. 4 Settling mechanisms Next we conside thee foms of the settling tems S u and S v. We wite S u = K S u u m, S v = K S v v m, (13) whee K > is a constant. Diffeent foms of the dimensionless settling functions S u and S v ae now discussed.
584 A. J. Tewenack et al. 4.1 Settling mechanisms independent of pheomone 4.1.1 Settling mechanism (i): constant settling ate The simplest settling mechanism is obtained by setting S u = S v = 1, coesponding to animals settling at constant ate K. Boadbent et al. [9], Okubo [24] and Williams [35] discuss the one-dimensional vesion of this settling mechanism combined with dispesal mechanism A (linea diffusion) in the context of insects. Shigesada [28] solves a adially-symmetic model with this settling tem and a population-pessue dispesal mechanism simila to dispesal mechanism B (see Sect. 6.2). 4.1.2 Settling mechanism (ii): population-dependent settling ate A moe inteesting case is whee the ate at which settling occus depends on the population density, that is S u and S v ae a function of the total population. We assume that the population pefeentially settles in less cowded aeas. A simple choice fo S u and S v is S u = F(1 u u ), S v = F(1 u + v u ), (14) whee u is a theshold population density, and F(w) is a continuous function that switches fom zeo to one in the neighbouhood of w =. In the numeical esults pesented in Sect. 7, we used a amp function defined as w w F(w) = h <w<h 1 w h. (15) Othe continuous functions can be used (fo example, a tanh-based function), which poduce qualitatively simila esults. The biological intepetation of such a settling function is that settling occus in egions whee the total population density is less than the theshold density u and no settling occus in egions whee the total population density is geate than u. The behaviou in egions whee the density is appoximately u depends on the choice of F(w). 4.2 Settling mechanisms dependent on pheomone We discuss one settling mechanism whee the settling ate is dependent upon the pheomone concentation.
Dispesal and settling of tanslocated populations 585 Table 2 Summay of the dimensionless settling functions S u and S v fo the diffeent settling mechanisms. In ou numeical esults, F( ) wasdefinedasin(15) Label Mechanism S u S v (i) Constant settling 1 1 ( (ii) Population-dependent settling F 1 u ) ( u F 1 ūs() ) + v u ( (iii) Pheomone-dependent settling F 1 c ) ( u c F 1 c ) u + c v c 4.2.1 Settling mechanism (iii): pheomone-dependent settling ate As fo the population-dependent settling ate, we conside a simple fom of S u and S v whee ( S u = F 1 c ) ( u c, S v = F 1 c ) u + c v c, (16) whee c is a theshold pheomone concentation. Again, the F(w) given in (15) was used to obtain ou numeical esults. In this case, animals will settle if the total pheomone concentation is less than c, and will not settle if the total pheomone concentation is geate than c. The behaviou when c c depends on the function chosen fo F(w). Table 2 gives a summay of settling mechanisms (i) (iii). Each combination of dispesal and settling mechanism constitutes a diffeent model, which we will efe to as A(i), A(ii), A(iii), B(i), etc. 5 Dimensionless model equations Hee we give the systems of equations fo the fist and second tanslocations in tems of dimensionless vaiables. Details of the scalings ae given in Appendix A. 5.1 The fist tanslocation The system of equations in tems of dimensionless vaiables and paametes fo the fist tanslocation is u m u s c u = µ 1 ( D u u ) m S u u m, (17) = S u u m, (18) = α 1 ( c ) u + β(u c u ), (19)
586 A. J. Tewenack et al. whee D u and S u ae given in Tables 1 and 2. Hee we have intoduced thee dimensionless paametes µ = D KL 2, α = D c KL 2, β = λ K, (2) whee L is a length scale which depends on the dispesal mechanism. The paametes α and µ ae dimensionless animal and pheomone motility espectively and β is the atio of the pheomone decay ate to animal settling ate. As thee is no epulsion-based dispesal fo the fist tanslocation, all dispesal mechanisms ae diffusion-based. Fo models A, B, C (i) (iii), we choose a diffusive length scale L = D/K so that µ = 1. Howeve, fo dispesal mechanism D, epulsion is the dominant dispesal mechanism in the second tanslocation. In this case we theefoe choose to scale both the fist and second tanslocations with the epulsion length scale, as descibed in Sect. 5.2 below. Note that α depends on ou choice of L, allowing us to specify the ate of diffusion of the pheomone elative to the dominant dispesal mechanism. This system of equations is solved with no-flux bounday conditions at, initial conditions (3) and c u (, ) = as the initial condition fo the pheomone. 5.2 The second tanslocation The equations fo the second tanslocation ae v m = µ 1 ( D v v ) m + γ 1 ( v m c ) u S v v m, (21) v s = S v v m, (22) c u = α 1 ( c ) u + β(u c u ), (23) c v = α 1 ( c ) v + β(v c v ), (24) whee D v and S v ae given in Tables 1 and 2. As discussed, hee we assume that the second tanslocation occus when the fist population has eached steady state, so that u =ū s (), the final settled population. The dimensionless paametes ae those given in (2), togethe with γ = ku χ λkl 2. (25) Fo dispesal mechanisms A C, we set L = D/K giving µ = 1, and since thee is no epulsion γ =. Fo dispesal mechanism D, we set L = ku χ/(k λ), so that γ = 1. This system of equations is solved with no-flux bounday conditions at and initial conditions (6). The initial conditions fo the two pheomone concentations ae
Dispesal and settling of tanslocated populations 587 c u (, ) = c u () and c v (, ) =, whee c u () is the steady state distibution of the pheomone poduced by the fist tanslocated population, coesponding to u =ū s (). We next look at solutions to the models fo both the fist and second tanslocations. Mostly numeical solutions ae pesented. Howeve, fo the fist tanslocation poblem, thee ae some simple cases fo which an exact o appoximate solution may be found analytically. We begin by discussing these cases. 6 Analytical esults 6.1 Solution to Model A(i): constant diffusivity with constant settling An analytic solution is obtained fo Model A(i), whee the diffusivity and the settling ate of the animals ae constants. Okubo [24] gave a solution fo the one-dimensional poblem. We conside the adially symmetic two-dimensional case, whee the fist tanslocation satisfies u m = 1 ( u ) m u m, (26) u s = u m. (27) Inteestingly, the steady states cannot be detemined by setting the left hand sides of (26) (27) to zeo. The only infomation we obtain is u m =. The steady state solution fo u s cannot be detemined a pioi. To find the steady state solution of u s,we must detemine the tansient solutions and obseve how they evolve in time. Using the solution to (26), we find the steady state solution ū s () by integating (27) ove time and taking the limit as t : ū s () = Making the substitution σ = e τ /2 yields ū s () = Q u 4π exp[ σ 2 4σ ]dt. (28) Q u 4π e cosh τ dτ = Q u 2π K (), (29) whee K is the modified Bessel function of the second kind of ode zeo [34]. This solution is also given by Shigesada [28] and is shown in Fig. 3. In this model, thee is no inteaction between animals eithe within o between the fist and second tanslocated populations. Theefoe the steady state solution fo the second elease is v s () = Q v K ()/2π. Two points can be obseved fom the distibution. Fistly, the model pedicts an infinite density of animals at the oigin, and secondly, the tails of the distibution extend to infinity. This is due to the constant diffusivity coefficient.
588 A. J. Tewenack et al. Fig. 3 The steady state solution to Model A(i) (constant diffusivity with constant settling) fo the fist elease with Q u = 1 6.2 An appoximate solution to Model B(i): population-dependent diffusivity with constant settling Combining dispesal mechanism B (population-dependent diffusivity) with settling mechanism (i) (constant settling ate) yields the system u m u s = 1 ( (u m + u s ) u ) m u m, (3) = u m. (31) Replacing u m + u s with u m only in the diffusivity, allows the fist equation to be decoupled fom the second as u m = 1 ( ) u m u m u m. (32) Following Shigesada [28], we make the substitutions τ = 1 e t and n m (,τ) = u m (, t)e t,giving n m τ = 1 ( ) n m n m. (33) Using the solution to (33) given by Pattle [25], and tansfoming back to the oiginal vaiables, the motile fist tanslocated subpopulation is given by ( ) 2Q u e t u m (, t) = π 1 2(t) 1 2 1 2(t), 1 (t), > 1 (t), (34) whee 1 (t) = ( ) 1/4 16Qu π (1 e t ). (35)
Dispesal and settling of tanslocated populations 589 Fig. 4 The steady state solution to Model B(i) (population-dependent diffusivity with constant settling) fo the fist elease with Q u = 1 The aea eventually inhabited by the animals has adius 1, whee 1 = lim t 1(t) = ( ) 1/4 16Qu. (36) Let t 1 () be the time that the population fist eaches distance fom the point of elease. It is given by t 1 () = ln 1 π 4. (37) 16Q u To obtain u s (, t), we integate u m (, t) fom time t 1 () to t. Thus u s (, t) = t t 1 () π u m (,σ)dσ, (38) 1 2(t) [ ( ) 2 ( ) 2 ( ) ] 1 + 2 ln, = 1 (t), 4 1 (t) 1 (t) 1 (t), > 1 (t). Letting t and eplacing 1 (t) with 1, we obtain ū s (), the steady state distibution of the population. In contast to the solution with a linea diffusivity in Sect. 6.1, the solution to this model is finite eveywhee and is esticted to a finite egion, as illustated in Fig. 4. Howeve, fo the nonlinea diffusion case, we cannot detemine an analytic appoximation to the second tanslocated population, whee the diffusivity is now ū s + v m + v s, since the inteaction tem ū s () cannot be ignoed. Numeical solutions to the oiginal model (3) (31) ae found to have a simila shape, width and height to the analytical solutions to the simplified equations, as discussed in Sect. 7.4. (39) 7 Numeical esults and discussion Systems (17) (19) and (21) (24) wee solved numeically using Matlab s pdepe solve on < < R fo t >. The value of R was chosen to be sufficiently lage so that the effect of the bounday was minimal. Theefoe R was chosen to be geate than
59 A. J. Tewenack et al. the adius of the settled egion fo those models whee the settled egion of the steady state solution was bounded, and sufficiently lage so that the population density at R was negligible othewise. The systems wee solved fo each combination of dispesal and settling mechanism, esulting in the twelve models A(i), A(ii),...,D(iii). Fo the diffusion-based models (A C), we set γ = and the system was scaled with espect to diffusivity, while fo the epulsion model (D), the system was scaled with the epulsive chemotactic mechanism, as detailed in Appendix A. 7.1 The fist tanslocation Figue 5 shows the steady state solutions ū s () fo each combination of dispesal and settling mechanism fo a given set of paamete values. Chemotaxis is not included as a dispesal mechanism fo the fist elease because thee ae no unfamilia animals and thus no unfamilia pheomone pesent when the fist population is eleased. As illustated in Fig. 5, the solutions vay between models. It is appaent that the distibutions fall into two classes. The fist consists of distibutions in column (i). These distibutions have a single peak at =, the point whee the animals wee eleased. The models in column (i) have a constant ate of settling, and theefoe the animals begin to settle immediately upon thei elease, esulting in the peak at =. Consequently, the egion aound the elease point is fa moe densely populated than suounding aeas. 2. i.25 ii.25 iii A B 2. 1 2 3 4 5.25 1 2 3 4 5.25 1 2 3 4 5 C 2. 1 2 3 4 5.25 1 2 3 4 5.25 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 Fig. 5 Steady state solution of ū s () vesus fo fist elease. Columns (i), (ii) and (iii): Settling mechanisms. Rows A, B, C: Dispesal mechanisms. Hee α =.1, β = 1, Q u = 1, u =.25, c =.25,.25.25 and t =.33. Note the vetical scale fo column (i) is diffeent fom those in (ii) and (iii)
Dispesal and settling of tanslocated populations 591 2. i.25 ii.25 iii A 2. B 1 2 3 4 5.25 1 2 3 4 5.25 1 2 3 4 5 2. C 1 2 3 4 5.25 1 2 3 4 5.25 1 2 3 4 5 2. D 1 2 3 4 5.25 1 2 3 4 5.25 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 Fig. 6 Steady state solutions ū s () (black line) and v s () (ed line) vesus. Columns (i), (ii) and (iii): Settling mechanisms. Rows A, B, C, D: Dispesal mechanisms. Hee α =.1, β = 1, u =.25, c =.25,.25.25 and t =.33 fo all mechanisms. Fo dispesal mechanisms A C, µ = 1andγ =. Fo dispesal mechanism D, µ =.1 andγ = 1 (the dimensionless length scale is chosen diffeently fom A C). Note the vetical scale fo column (i) is diffeent fom those in (ii) and (iii) The second class of distibutions consists of those in columns (ii) and (iii). While some have a small peak at =, thee is a significant peak away fom the oigin. Fo these models, settling begins on the outskits of the dispesing population, since eithe the population density o pheomone concentation at the point of elease emains high fo some time. Howeve in column (iii) (pheomone-dependent settling) we obseve small peaks at =. Hee animals begin to settle at the elease site befoe the pheomone concentation builds up to the theshold level c. 7.2 The second tanslocation The ed distibutions in Fig. 6 ae the steady state solutions fo the second elease fo ou chosen set of paamete values. Again the second elease gives a peaked distibution, with the peak away fom the elease site, fo settling mechanisms (ii) (iii). Models B D(ii) and B D(iii) yield steady state solutions that ae qualitatively simila, whee the two populations inhabit quite diffeent egions. The second population fills the
592 A. J. Tewenack et al. inne egion whee the fist elease population is at lowe density, befoe dispesing futhe away to avoid the peaks of high density in the fist elease population. The egion is thus elatively evenly populated oveall, with the distibution extending ove a finite aea. Howeve, the two populations ae somewhat segegated, with the majoity of the second population foming a ing aound the fist population. The segegation is not as ponounced in the steady state distibutions of Models A(ii) (iii). These distibutions diffe slightly because the linea dispesal mechanism allows the second population to spead futhe befoe settling. As a esult, the distibutions of the fist and second populations ae simila, although the peak in density is futhe away fom the elease point fo the second population. Thee is little diffeence between the distibution of the fist population and the distibution of the second population fo Models A C(i). Fo Model A(i), the dispesal and settling mechanisms ae both independent of u, v and c; hence the pesence of the fist population does not affect the dispesal and settling of the second. Fo Models B C(i), the settling mechanism is again independent of u, v and c. Howeve the dispesal mechanisms depend on the population and the pheomone espectively. Fo this set of paamete values, Model D(i) has a unique featue both the fist and second populations have a peak in density at =, and the second population also has a peak at some = o. The dispesal speed of animals in the second population vaies moe noticeably with position due to the epulsive flux mechanism. In egions of slowe dispesal, moe settling occus, esulting in the lage peak at the elease site and the smalle peak aound =.6. This model pedicts a cental egion densely inhabited by both populations, with an oute egion pedominantly inhabited by the second population. 7.3 Choice of paamete values The values of Q u and Q v used in Figs. 5 and 6 ae appopiate to the case study discussed in Sect. 8. The emaining dimensionless paamete values wee chosen so that the esulting steady state solutions diffeed noticeably between models. Some obsevations can be made egading the behaviou of models fo cetain paamete values. The paametes α and β affect the poduction, decay and diffusion of the pheomone. Lage values of α esult in the pheomone diffusing quickly elative to the movement of the animals. To investigate a situation whee the pheomone pesists long enough fo the animals to espond to its pesence, we chose to conside a small value of α in ou numeical esults. Similaly, small values of β esult in the pheomone being poduced slowly elative to the time scale on which the dispesal is occuing. We ae inteested in cases whee the concentation of pheomone pesent is sufficient to affect the movement of the animals, and thus chose a elatively lage value fo β. Fo cetain values of α and β, flux mechanisms B and C ae vey simila. The paametes α and β may be chosen (α sufficiently small and β sufficiently lage) so that the pheomone is poduced and decays quite apidly, in which case the pheomone concentation closely esembles the animal population. In this case, flux mechanism C is simila to flux mechanism B. Howeve if the paametes ae chosen so that the
Dispesal and settling of tanslocated populations 593 pheomone decays moe slowly, and eflects the population density at ealie times, the two flux mechanisms ae moe distinct. The choice of theshold densities of u and c has a significant effect on the steady state distibutions of the applicable models. If these ae chosen to be high elative to the numbe of animals pesent, they will have little effect on the settling pocess. We chose theshold densities that wee lowe than the maximum density expected if no theshold existed, thus ensuing that steady state solutions to models with thesholds would diffe fom those without thesholds. The value h in equation (15)wassettoa faction of the theshold values, as h =.156. 7.4 Size of the settled egion fo the fist tanslocation Befoe a tanslocation is caied out, it would be helpful to have an estimate of the aea of suitable habitat equied to tanslocate a cetain numbe of animals. In addition, an estimate of the seach aea would help in monitoing a tanslocated population. Theefoe, a pediction of the distance that tanslocated animals ae likely to dispese would be useful to zoologists and wildlife manages. We can obtain an estimate of the maximum distance 1 that animals in the fist tanslocation dispese fom the point of elease, using ou numeical esults. A compaison of 1 as a function of the size of the population Q u fo each of the models is illustated in Fig.7, togethe with distances obtained using the analytic esults in Sects. 6.1 and 6.2. When ū s () is infinite in extent fo some models, we define 1 as the position whee ū s ( 1 ) =.1. Fo settling mechanism (i), the analytic esults ae in good ageement with the numeical esults. The analytic esults fo Model B(i) detemine the adius to incease with Qu 1/4. We obseve that the behaviou of 1 as Q inceases is simila fo all othe models 1 inceases with deceasing slope as Q gets lage. 7.5 Size of the settled egion fo the two tanslocations In designing a tanslocation, it would be useful to know how the final distibution of animals diffes if the animals wee eleased in two goups some time apat, athe than in a single tanslocation. Figue 8 compaes the total population density afte the second elease has settled, namely ū s () + v s (), and the steady state distibution that esults if the entie goup of animals Q u + Q v was eleased in one single elease. Note that since thee is no epulsion in the fist elease, the single tanslocation in ow D coesponds to constant diffusivity with µ =.1. Note that each adially symmetic distibution epesents the same numbe of animals. Inteestingly, the esults in the figue suggest that the size of the settled aea diffes vey little between the single and double tanslocations. Fo settling mechanisms (ii) and (iii), a double tanslocation esults in a moe unifomly populated aea, as thee is no significant dip in the total density aound the point of elease. Fo these settling mechanisms, animals in the second elease have the oppotunity to settle in the less densely populated egion aound the point of elease.
594 A. J. Tewenack et al. - 1 3.5 3 2.5 2 1.5 1.5 Settling mechanism (i) (constant settling ate).5 1 1.5 2 2.5.5 1 1.5 2 2.5 Q u Q u - 1 3.5 3 2.5 2 1.5 1.5 Settling mechanism (ii) (population dependent settling ate) 3.5 3 2.5 2-1 1.5 1.5 Settling mechanism (iii) (pheomone dependent settling ate).5 1 1.5 2 2.5 Q u Fig. 7 Compaison of the (scaled) adius of the settled egion of the fist tanslocation 1 vesus the (scaled) numbe of animals eleased Q u fo diffeent mechanisms. Numeical esults fo Model A (open cicles), Model B ( ) and model C (+) ae given. The analytical esults fo model A(i) ae Q u K ( 1 )/2π =.1 (dashed line) and fo model B(i) ae 1 (Q u ) = (16Q u /π) 1/4 (dash-dot line) Results fo settling mechanism (i) in Fig. 8 diffe fom the othe two settling mechanisms. Fo Model A(i), the single and double tanslocations esult in identical distibutions, since the pesence of the fist population has no influence on the second population. Fo Models B C(i), the steady state distibution afte the single tanslocation has a slightly lage settled egion, while the opposite occus fo Model D(i). The epulsion of the second tanslocated population fo Model D(i) motivates the second population to quickly move away fom the fist population. This effect is not as significant fo Models D(ii) (iii) because the gadient of the pheomone distibution is now positive nea the point of elease, athe than negative as in Model D(i). Yet even fo Model D(i), the size of the settled egion fo the single tanslocation is a good estimate fo the size of the settled egion fo the double tanslocation. 7.6 Effect of a finite domain So fa, in ou esults and discussion, we have assumed an infinite domain. This poblem is useful fo estimating how much space would be desiable to tanslocate a population of a given size, o to suggest how fa a tanslocated population may spead. Howeve, if tanslocating a population into an aea that is esticted in size fo some eason, it may be useful to solve the poblem on a finite domain with no-flux bounday conditions. Fo example, it would be useful to know whethe all individuals ae likely to settle
Dispesal and settling of tanslocated populations 595 A i 2 1.5 1.5 1 2 3 4 5 2 1.5 B 1.5 1 2 3 4 5 2 1.5 C 1.5 1 2 3 4 5 2 1.5 D 1.5 1 2 3 4 5.2.15.1.5 1 2 3 4 5.2.15.1.5 1 2 3 4 5.2.15.1.5 1 2 3 4 5.2.15.1.5 1 2 3 4 5 ii.2.15.1.5.2.15.1.5.2.15.1.5 iii 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5.2.15.1.5 1 2 3 4 5 Fig. 8 Compaison of a single and a double tanslocation. Blue line: ū s () afte a single tanslocation with Q u = 2.33. Black line: ū s () + v s () afte a double tanslocation with Q u = 1andQ v = 1.33. Columns (i), (ii) and (iii): Settling mechanisms. Rows A, B, C, D: Dispesal mechanisms. Hee α =.1, β = 1, u =.25, c =.25,.25.25 and t =.33 fo all mechanisms. Fo dispesal mechanisms A C, µ = 1andγ =. Fo dispesal mechanism D, µ =.1 andγ = 1 (the dimensionless length scale is chosen diffeently fom A C). Note the vetical scale fo column (i) is diffeent fom those in (ii) and (iii) given the size of the tanslocated population and the size of the domain. Altenatively, we may wish to be able to distinguish between models using data collected fom a population whee the animals ae esticted to a smalle domain. Imposing a domain of finite size intoduces a new length scale into the poblem, so hee we limit ou discussion to some geneal obsevations and an example. Fist of all, we noted ealie that the steady state distibutions fo dispesal mechanisms B and C ae finite in extent, while those fo mechanisms A and D ae infinite in extent. Consequently, imposing a bounday beyond the settled egion fo dispesal mechanisms B and C does not change the steady state distibution. Fo mechanisms A and D, imposing a finite domain has little effect on the steady state distibution if the new bounday is placed sufficiently fa fom the point of elease. Fo example, efeing to Fig. 6, imposing no-flux bounday conditions at = 4 has little effect on distibutions in ows A and D, and no effect on distibutions in ows B and C. If the domain is esticted futhe, it is possible that some individuals in eithe o both eleases may not settle, depending on the settling mechanism. Fo settling
596 A. J. Tewenack et al. 2. i.25 ii.25 iii A.5 1 1.5 2. B.5 1 1.5.25.25.5 1 1.5 2. C.5 1 1.5.25.5 1 1.5.25.5 1 1.5 2. D.5 1 1.5.25.5 1 1.5.25.5 1 1.5.5 1 1.5.5 1 1.5.5 1 1.5 Fig. 9 Steady state solutions ū s () (black line), v s () (solid ed line) and v m () (dashed ed line) vesus with no-flux bounday conditions at = 1.5. Columns (i), (ii) and (iii): Settling mechanisms. Rows A, B, C, D: Dispesal mechanisms. Hee α =.1, β = 1, u =.25, c =.25,.25.25 and t =.33 fo all mechanisms. Fo dispesal mechanisms A C, µ = 1andγ =. Fo dispesal mechanism D, µ =.1 andγ = 1 (the dimensionless length scale is chosen diffeently fom A C). Note the vetical scale fo column (i) is diffeent fom those in (ii) and (iii) mechanism (i), this will not occu, as individuals will eventually settle egadless of thei local envionment. Howeve fo settling mechanisms (ii) and (iii), if the domain is too small, the system will each a steady state whee u m and/o v m ae not equal to zeo eveywhee. The net flux of the motile population is zeo eveywhee. Howeve, individual animals may move about povided that the net flux is zeo eveywhee. Thus we have a population that has the potential to be motile but is unable to move. If, fo example, the domain inceased in size o the theshold density inceased, this population would then be able to dispese futhe and settle if the new conditions allowed it. As an example, Fig. 9 shows the steady state solutions obtained by solving the system with no-flux bounday conditions at = 1.5. (This value povides a typical illustative example of the inteaction between the tanslocated populations and the boundaies of the finite domain.) This figue should be compaed with Fig. 6, which shows the steady state solutions to the same poblem on an infinite domain. Fo models with settling mechanism (i), both populations settle completely, as expected. The steady state motile population v m () is zeo eveywhee. These distibutions diffe
Dispesal and settling of tanslocated populations 597 vey little fom the coesponding distibutions in Fig. 6, as the population density beyond = 1.5 is vey small on the lage domain. Thus only a small popotion of the population is affected by a bounday at = 1.5. Fo all of the models A(i) to D(iii), the fist population successfully settles completely; that is, ū m () = eveywhee. Howeve, fo settling mechanisms (ii) and (iii), we obseve that the system eaches a steady-state distibution whee not all individuals in the second tanslocation settle. The population that has not settled, v m (),isshown in Fig. 9. Fo the diffusion-based dispesal mechanisms A C, the motile population is spead at unifom density acoss the domain. Howeve, the combination of chemotaxis and a small amount of diffusion in dispesal mechanism D causes a non-unifom density v m () at steady state. We obseve in the figue that fo settling mechanism (ii), the animals do not colonise the domain up to the theshold density. This is because the domain becomes cowded quite quickly afte the second tanslocation. Once the second population has dispesed, thee ae no aeas whee the total population density is low enough fo any individuals to settle. Settling only occus at the oute egion of the domain because the fist few animals to aive thee fom the second tanslocation see a low enough population density. One diffeence we can obseve between the distibutions shown in Fig. 9 is the popotion of animals that settle. This depends on the density of the fist population at the edge of the domain. As we have just discussed, the only oppotunity fo the second population to settle is at the edge of the domain befoe the second population has dispesed vey fa. Thus fo models A(ii) and A(iii), a vey small popotion of the second elease settles because the density of the fist population is elatively high nea the bounday. Fo models B, C, D (ii) (iii) howeve, the population density afte the fist elease is low at the edge of the domain, poviding an oppotunity fo some individuals in the second population to settle. Thus these models have a geate popotion of settled individuals in the second elease. The diffeences between settling mechanisms in this example highlights the impotance of identifying the appopiate mechanisms. 7.7 Two diffeent elease sites In the case study that motivated this wok (Sect. 8), the two tanslocated populations wee eleased at the same location. Howeve, fo many double tanslocations this may not be feasible o desiable. Of couse the poblem is then no longe adially symmetic, and so analytical esults ae not possible. Futhemoe, the steady state distibutions will depend on the distance between the two points of elease and the population sizes. We will discuss an example in one dimension, but ou conclusions will also be elevant to the two-dimensional poblem. Figue 1 illustates the steady state distibutions obtained by solving the onedimensional equivalents of systems (17) (19) and (21) (24) fo model B(ii). The figue demonstates the elationship between the diffeent length scales pesent in this poblem, namely the distance between the two elease points, and the widths of the settled population fom single tanslocations of size Q u and Q v. If the distance
598 A. J. Tewenack et al. a b c.2.2.2.1.1.1 1 5 5 1 x 1 5 5 1 x 1 5 5 1 x Fig. 1 Steady state distibutions esulting fom two diffeent elease sites fo model B(ii): one-dimensional poblem. Steady state solutions ū s (x) (black line) and v s (x) (solid ed line) vesusx. a Both populations eleased at x =. b Populations eleased at x =±5. c Populations eleased at x =±2. Hee u =.25, x =.25, t =.25, µ = 1, γ =, Q u = 1andQ v = 1.33 between the two elease points is lage than a citical distance, we will expect minimal inteaction between the two populations. The citical distance is the sum of the halfwidths of each single tanslocation, which is a function of Q u and Q v. In paticula, fo those models with a finite settled width such as B(ii), we can expect no inteaction at all between the two populations. Figue 1b is an example of a situation whee the second population is placed sufficiently fa fom the fist population so that thee is no inteaction between the two populations. Howeve, if the second population is eleased at a distance that is less than the citical distance (Fig. 1c), then the pesence of the fist population will affect the dispesal and/o settling of the second population since the two populations inteact. (The exception to this is model A(i), whee thee is no inteaction at all between the two populations.) We obseve that the steady state distibution of the second population is not symmetic, and that some animals on the left hand side of the distibution have had to tavel though the egion inhabited by the fist population in ode to find somewhee to settle. Figue 1a is symmetic about x = as the two populations wee eleased at the same point, as fo ou peviously pesented esults. The decision of how to aange a double tanslocation depends on the equiements and limitations of the tanslocation pogamme. Fo example, intoducing the second population fa fom the fist, as in Fig. 1b, equies animals to tavel the shotest distance befoe settling. Howeve, moe space is equied, and thus moe esouces ae equied to monito the population. Next we show how this wok can be applied to the paticula case of the Maud Island fog. 8 Case study The tial tanslocation on Maud Island was conducted in 1984 and 1985 [4]. Initially, 43 fogs wee eleased at the new site, followed by anothe 57 fogs at the same position 12 months late. It was obseved that no fogs wee pesent pio to the fist tanslocation. Seaches of the aea wee conducted at least annually. When a fog was discoveed, it was identified and its position was ecoded as gid coodinates to the neaest.5 m. A list of obseved locations was compiled fo each fog.