Corrigendum to “Non-local interactions and the dynamics of dispersal in immature insects” [J. Theor. Biol. 185 (4) (1997) 523–531]

Journal of Theoretical Biology 336 (2013) 252–258

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Corrigendum

Corrigendum to “Non-local interactions and the dynamics of dispersal in immature insects” [J. Theor. Biol. 185 (4) (1997) 523–531] J.L. Boldrini a, R.C. Bassanezi a, A.C. Moretti a, F.J. Von Zuben c, W.A.C. Godoy d, C.J. Von Zuben e, S.F. dos Reis b,n a

Departamento de Matemática Aplicada, Universidade Estadual de Campinas, 13083-970 Campinas, São Paulo, Brazil Departamento de Parasitologia, Universidade Estadual de Campinas, 13083-970 Campinas, São Paulo, Brazil c Departamento de Computação e Automação Industrial, Universidade Estadual de Campinas, 13083-970 Campinas, São Paulo, Brazil d Departamento de Parasitologia, Universidade Estadual Paulista, 18618-000 Botucatu, São Paulo, Brazil e Departamento de Zoologia, Universidade de Brasilia, 70910-900 Brasilia, DF, Brazil b

art ic l e i nf o

a b s t r a c t

Available online 2 August 2013

A simple mathematical model is developed to explain the appearance of oscillations in the dispersal of larvae from the food source in experimental populations of certain species of blowflies. The life history of the immature stage in these flies, and in a number of other insects, is a system with two populations, one of larvae dispersing on the soil and the other of larvae that burrow in the soil to pupate. The observed oscillations in the horizontal distribution of buried pupae at the end of the dispersal process are hypothesized to be a consequence of larval crowding at a given point in the pupation substrate. It is assumed that dispersing larvae are capable of perceiving variations in density of larvae buried at a given point in the substrate of pupation, and that pupal density may influence pupation of dispersing larvae. The assumed interaction between dispersing larvae and the larvae that are burrowing to pupate is modeled using the concept of non-local effects. Numerical solutions of integro-partial differential equations developed to model density-dependent immature dispersal demonstrate that variation in the parameter that governs the non-local interaction between dispersing and buried larvae induces oscillations in the final horizontal distribution of pupae. & 2013 Elsevier Ltd. All rights reserved.

1. Introduction Dispersal is one of the most significant and complex aspects in the life history of organisms, both in its implications for the dynamics of populations (Roughgarden et al., 1988; Rohani and Miramontes, 1995) and in the genetic changes associated with movement and breeding (Gilpin, 1991; Nagylaki, 1992; Felsenstein, 1994). The complexity of dispersal is further enhanced by the spatial scale in which it occurs, which can vary from centimeters to hundreds of kilometers (Gaines and Bertness, 1993). In the case of immature organisms in particular, Gaines and Bertness (1993) identified three factors that determine the distribution of propagules following dispersal, viz., the density and distribution of reproductive adults, the timing and magnitude of their reproductive output, and the probability distribution of juvenile transport distances following release by an adult.

n

DOI of original article: http://dx.doi.org/10.1006/jtbi.1996.0320 Corresponding author. E-mail address: [email protected] (S.F. dos Reis).

0022-5193/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jtbi.2013.07.020

Whereas these three main factors were considered by Gaines and Bertness (1993) primarily in the context of dispersal of immatures mediated by fluid flows, these phenomena are likely to be of significance for the dispersal of juveniles in other organisms as well. For example, in the case of dispersal in insects whose immature forms live in ephemeral and discrete patches, the first two factors mentioned above interact to determine the density of juveniles and the structure of the community (Hanski, 1987), which will in turn probably affect the distribution of juveniles after their dispersal (de Jong, 1979; Ives, 1989, 1991). For insects whose immature forms live in ephemeral and discrete patches such as, for example, calliphorids, sarcophagids and drosophilids, two phases in the life-history are of paramount significance for the fitness of adults. During the first phase, the larvae that hatch from eggs usually experience crowding which results in exploitative competition for resources at a time when the larvae need to acquire the minimum amount of food necessary for successful pupation (Levot et al., 1979; Mueller, 1986, 1988). Density-dependent competition among larvae at this stage adversely affects fitness traits, such as fecundity and survival, and results in a delayed effect on the adults with consequences for

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the population dynamics (Prout and McChesney, 1985; Godoy et al., 1993; Von Zuben et al., 1993). In the second phase, the postfeeding larvae that have already attained the minimum weight necessary for pupation begin dispersing outside the food source in search of suitable sites for pupation, and the form and rate of this process will be a function of the probability distribution of distances traveled by juveniles. We are now left with the question of what are the factors affecting the probability distribution of dispersing juveniles. In this context there is increasing evidence for a relationship between density-dependence (crowding) and dispersal in insects (Taylor and Taylor, 1977; Taylor, 1978; Okubo, 1980; Bengtsson et al., 1994), which in turn may exert a significant effect on pupation (Tschinkel and Willson, 1971; Yto, 1977; Nakakita, 1982). The mechanisms by which larvae perceive crowding and how this relates to pupation are not well understood. Recently, however, Kotaki and Fujii (1995) demonstrated that crowding does inhibit pupation in the tenebrionid beetle Tribolium freemani, and that both mechanical and chemical stimuli may be involved in the perception of larval density in this species. In this note we approach the problem of dispersal in immature insects developing in discrete and ephemeral substrates where the spatial scale of dispersal is very small, of the order of a few meters. This note was originally motivated by an attempt to understand the results of Godoy et al. (1995) who studied dispersal in experimental populations of three calliphorid blowflies; Cochliomyia macellaria, a species native to the Americas and Chrysomya putoria and Chrysomya megacephala, two species that have been introduced from the Old World and Australasia (Guimarães et al., 1978). In the experiment reported by Godoy et al. (1995), larval dispersal from the food source was measured as the distance traveled by the larvae until they burrowed into the substrate to pupate. The final pattern of larval dispersal was represented as the horizontal frequency distribution of pupae recovered from the substrate. Larvae dispersed similar distances from the food source, although the three species differ strikingly in the form of dispersal (Fig. 1). In the native species, C. macellaria, the horizontal distribution of pupae decayed monotonically, whereas the two invading species, C. putoria and C. megacephala, showed oscillatory decaying distributions of pupae (Fig. 1). We remark that Von Zuben et al. (1996) developed an ad hoc model and obtained an expression that could be used to fit these data using nonlinear regression.

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Our primary objective here is to develop a framework to address the possible mechanism underlying the appearance of waves in the dispersal from the food source of immature blowflies. In these organisms the immature stage develops in the food source and the postfeeding larvae migrate from the food source to find a site to pupate. This life history can be characterized as a system with two populations; one composed of larvae actively migrating on the soil and the other composed of larvae that are burrowing into the soil to pupate. As an overall approach to understand the process of larval dispersal we conjecture that dispersal is influenced by larval density and that density can have an effect on pupation. In taking this line of reasoning, we rely on the available evidence for the interaction between density and the mechanical and chemical mechanisms associated with pupation (cf. Tschinkel and Willson, 1971; Nakakita, 1982; Kotaki and Fujii, 1995). In particular, we surmise that the larvae already buried in a point in the soil affect the burrowing capacity of the larvae above the soil in the neighborhood of that point, and consequently their pupation capacity. In other words, we assume that somehow the larvae above the soil sense when a certain spot is too crowded with buried larvae and, with a certain probability, avoid such a spot by moving and dispersing away in search for a less

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Fig. 1. Histogram of the experimental data for the number of buried pupae as a function of distance traveled as a result of larval dispersal from the food source in three blowfly species. (a) Chrysomya megacephala; (b) Chrysomia putoria; (c) Cochliomyia macellaria.

crowded spot. Obviously, the probability of avoidance and the neighborhood influence may depend on the species considered. The inhibition of pupation in a given neighborhood could conceivably

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be achieved by chemical and mechanical stimuli as demonstrated experimentally for tenebrionid beetles by Kotaki and Fujii (1995). We approach the problem posed by the appearance of waves in the dispersal of immature blowflies using the concept of non-local effects discussed in detail by Murray (1990, chapter 9). Non-local effects seem to arise naturally in many biological situations involving variations in density in a spatial setting. For example, during embryonic development non-local effects arise as a result of the high density of cells whose long filopodia are able to perceive variations in cell densities beyond their nearest neighbors (Murray, 1990, p. 530). Mesenchymal morphogenesis is, in particular, an example of a complex interaction between the motile cells and the elastic extracellular matrix (ECM). In this case nonlocal effects arise because the elastic fibers of the ECM can transmit stress between points that are far apart (Murray, 1990, p. 535). Non-local effects can also arise in the process of neural firing where the firing of nerve cells may be determined by excitatory and inhibitory input from cells that are not nearest neighbors (Murray, 1990, p. 481). Modeling biological phenomena in this context requires the identification of phenomena operating at non-local spatial scales, and the mathematical formalism can be developed from integral equations leading to integro-differential equations (Murray, 1990). In general terms, the process of modeling non-local interactions can be explained simply as follows (Murray, 1990, p. 246). Suppose that the rate of change in density, nðx; tÞ, at position x at a time t depends on the influence of nðx′; tÞ at all other positions in space, denoted by x′. A model in one space dimension that quantifies the effect that the neighboring nðx′; tÞ exert on nðx; tÞ can be written as Z 1 ∂n ðx; tÞ ¼ f ðnÞ þ wðxx′Þnðx′; tÞdx′: ð1Þ ∂t 1 In this equation wðxx′Þ is known as the kernel function that quantifies the non-local interaction in the space variable, and f ðnÞ is the usual source term well known in the reaction diffusion approach. The kernel, w, is the most important term in the integral equation, and its specification depends on the particular biological problem at hand (Murray, 1990, p. 247). In what follows we will show that the inclusion of a non-local interaction in the mathematical model is enough to explain, at least qualitatively, the appearance of oscillations in dispersal of the blowflies analyzed experimentally by Godoy et al. (1995). The mathematical model is presented at the physics level of rigor as is customary in theoretical ecology (cf. Ginzburg, 1983, p. 8), and consequently we assume without proof the existence and uniqueness of the solutions of the differential equations considered. A detailed mathematical analysis of the model demonstrating results on existence and uniqueness of solutions and their asymptotic behavior in time can be found in Boldrini and Moretti (1995). We will describe briefly these results just after mathematically defining the model. Our intention here is to understand in a qualitative way some of the main features of the observed results of dispersal, and, in particular, to raise a plausible explanation for the observed oscillatory patterns in the final horizontal frequency distribution of buried blowfly pupae. For simplicity, and also to make comparisons with the results of actual laboratory experiments, we will describe a model in onedimension in space. The postfeeding behavior of blowfly larvae will be modeled by splitting the total larval population in two parts; the larvae dispersing above the soil and the larvae that have already buried to become pupae. Based on balance (conservation) principles, we will describe equations governing the changes with time and distance in these subpopulations. We will consider the initial time (zero) to be the time at which the larvae start leaving the food substrate. As in the experimental setting (Godoy et al., 1995), the larvae can only move to the right of the food, which is assumed to be the origin of the axis of reference.

Thus, we will look for functions u and v whose biological meanings are the following:

 uðx; tÞ gives the density of dispersing larvae above soil at distance x Z0, at time tZ0;

 vðx; tÞ gives the density of buried larvae at distance xZ0, at time tZ0. We denote by Jðx; tÞ and Fðx; tÞ, respectively, the values at ðx; tÞ of the horizontal flux of the moving larvae and their burying rate. We also observe that, once buried, a larva can no longer move horizontally, and thus u and v at a point ðx; tÞ, with x 40 and t 40, must satisfy the following balance equations: 8 ∂u > > < ðx; tÞ þ divJðx; tÞ ¼ Fðx; tÞ; ∂t ð2Þ ∂v > > : ðx; tÞ ¼ Fðx; tÞ: ∂t Here, div denotes the divergence operator, which in one dimension reduces to ∂=∂x. We also assume that the moving larvae have a constant mean speed V 0 to the right (this is not essential, it is just to simplify the subsequent analysis). This seems to be according to observation and experimental evidence: it is observed that as soon as the larvae leave the food source, they move away from it at a definite positive mean velocity, which appears not to be just a consequence of random motion (Greenberg, 1990). This, however, must be further studied, and here we take the simpler view in order to devise a reasonable explanation for the appearance of oscillation in the distribution of buried larvae. Thus, at this level of simplicity, the flux Jðx; tÞ can be taken similar to an advection flux: Jðx; tÞ ¼ V 0 uðx; tÞ: ð3Þ Other mathematical possibilities, like a random diffusion mechanism, could also be added to or even substitute for the above J, but they would not change the essential mechanism of production of oscillatory patterns, which, as we will see, will depend just on the coupling between some sort of transport (movement) and a nonlocal interaction with saturation. As for the burying rate F(x, t), according to what was previously said, we will assume Fðx; tÞ ¼ Bðvð U ÞÞðx; tÞuðx; tÞ;

ð4Þ

that is, Fðx; tÞ is linearly proportional to the density uðx; tÞ of dispersing larvae at ðx; tÞ, with a proportionality factor that may depend on the entire distribution of the buried larvae vð U Þ, and not necessarily only on the value vðx; tÞ of v at that ðx; tÞ. Actually, we will assume the proportionality factor B, to be of the following special form: Z 1 Bðvð U ÞÞðx; tÞ ¼ maxf0; b0  KðsxÞvðs; tÞdsg: ð5Þ 0

Here, b0 4 0 is the basic burying rate factor and KðzÞ Z0 is a given function that is associated with the decrease of the effective burying rate at a point in the soil caused by the larvae buried in a neighborhood of that point. Note that KðzÞ here is the kernel function and the above integral is responsible for the non-local interaction of the two subpopulations, that is, the larvae actively dispersing above the soil and larvae that have buried to pupate. We have to have the R1 maximum between zero and b0 4  0 KðsxÞvðs; tÞds so that the burying rate is always non-negative, because we assume that a larva that has already buried in the soil cannot unearth itself and start dispersing again. This means that we have a saturation effect in our model, that is, if a large number of larvae are buried in a certain neighborhood then the effective burying rate near this neighborhood becomes zero, forcing the other larvae that have not yet buried to move to find another suitable place to bury in the soil. This argument is clearer in the important special case when the kernel KðzÞ is equal to the function H λ;a ðzÞ depending on two

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parameters λ Z 0 and a Z 0 and given by the following: take a fixed non-negative smooth function ΦðzÞ with support [  1, 1] that could be similar to the “hat-function” or any other function of this sort, and consider H λ;a ðzÞ ¼ ðλ=aÞΦðz=aÞ: In this case B is reduced to ( ) R xþa 0; b0 ðλ=aÞ maxfxa;0g Bðvð U ÞÞðx; tÞ ¼ max ; ΦððsxÞ=aÞvðs; tÞds

ð6Þ

ð7Þ

and it is clear that a gives the maximum distance of influence of the buried larvae to the reduction in the effective burying rate, while l is associated with the effectiveness of this influence (measuring the effectiveness of the response to chemical cues, for instance). Obviously, there are other possibilities for the influence function K. Other aspects to be considered are those concerning the required initial and boundary conditions. Since at time t ¼0 there are no larvae away from the food substrate, the initial conditions must be identically zero for both u and v. As for the boundary condition, we observe that the larvae leave the food substrate at x ¼0 and enter the domain x 40 at a certain rate that may depend on time (and in reality this rate becomes zero after a certain period of time, when all the larvae have left the food substrate). Thus, we will assume that we are given a function gðtÞ Z 0 that furnishes the rate of entrance of larvae in the domain x 40 (which is the same as the rate of leaving the food substrate). Now, we can put all the above information together to obtain the following boundary-initial value problem: 8 ∂u > > ðx; tÞ þ ∂ðV∂x0 uÞ ðx; tÞ ¼ > > ∂t >   > R1 > > max 0; b0  0 KðsxÞvðs; tÞds uðx; tÞ; > > > > ∂v < ðx; tÞ ¼ ð8Þ ∂t  >  R1 > > > max 0; b0  0 KðsxÞvðs; tÞds uðx; tÞ; > > > > uðx; 0Þ ¼ 0; vðx; 0Þ ¼ 0; 8 x 40; > > > > : uð0; tÞ ¼ gðtÞ; 8 t 4 0:

255

lowering the amplitude of the outer rings of local maximum concentrations due to angular dispersion). In this paper we decided to model in a simpler way to conform to the laboratory experiments in which the larvae are allowed to move in only one direction. (iii) Another possible alternative to model larval dispersal is to consider diffusion instead of advection. In this case the advection term in (8) would be replaced by a term D∂2 u=∂x2 . This point will be further discussed later on. In order to test the circumstances for which there are oscillatory patterns in the final distribution of buried larvae ðvð UÞÞ, we performed numerical experiments. We considered the situation expressed by Eqs. (4) and (7), and then the equations corresponding to Eq. (8) are 8 ∂u ðx; tÞ þ ∂ðV∂x0 uÞ ðx; tÞ ¼ > > > ∂t 8 9 > > 0; b ðλ=aÞ > > > > < R 0xþa = > > > >  maxfxa;0g ΦððsxÞ=aÞ uðx; tÞ; max > > > > > : ; > > vðs; tÞds > > < ∂v ð9Þ ∂t ðx; tÞ ¼ > ( ) R xþa > > > 0; b0 ðλ=aÞ maxfxa;0g > > > uðx; tÞ; > max > ΦððsxÞ=aÞvðs; tÞds > > > > > > uðx; 0Þ ¼ 0; vðx; 0Þ ¼ 0; 8 x 4 0; > > > : uð0; tÞ ¼ gðtÞ; 8 t 40: For the numerical experiments we will take ΦðzÞ as the continuous C 1 by parts function defined as zero if jzj 4 1, one if jzj o 0:9, and linear on the intervals 1 o z o 0:9 and 0:9 oz o 1. This kernel is under the conditions that guarantees the existence and uniqueness of solutions, as described in the Remarks after Eq. (8). To find the relevant combinations of biological parameters, we proceed with the non-dimensionalization of Eq. (9). For this, we define: g m ¼ maxfgðtÞ; t Z 0g 4 0; u^ ¼ u=g m ; v^ ¼ v=g m ;

2.1. Remarks (i) Boldrini and Moretti (1995) studied the above Eq. (8) from the mathematical point of view. They assumed that the kernel K satisfies K A L1 ð1; 1Þ \ Lp ð1; 1Þ for some 1 o p r þ 1 and K′ A L1 ð1; 1Þ (with the derivative K′ taken in the sense of distribution); gð U Þ is assumed to have compact support. With these hypotheses, by using semi-group theory and a priori estimates, they were able to prove that there is a unique global in time strong solution to the above equations [with u; v A C 1 ðð0; 1Þ; Lq ð0; 1ÞÞ \ Cðð0; 1Þ; W 1;q ð0; 1ÞÞ, where q is such that either 1=q þ 1=p ¼ 1 when 1 o p o þ 1, or any finite number larger than one when p ¼ þ 1]. Also, they proved that the above solution has well defined asymptotic limits as t- þ 1, namely, uð U ; tÞ-0 in Lq ð0; 1Þ and there is R1 R1 va A L1 ð0; 1Þ satisfying such that 0 va ðxÞdx ¼ 0 gðsÞds 1 vð U; tÞ-va ð U Þ in L ð0; 1Þ as t- þ 1. (ii) A possible alternative explanation to dispersing would be to model the dispersing larvae as a function of radial distance from a food source. In this case, the advection term in (8) changes to ð1=rÞð∂ðrV 0 uÞ=∂rÞ. This is a more natural modeling of the situation found in nature. From the qualitative point of view, however, the results concerning the presence of oscillations in the final distribution of buried larvae are exactly the same as the ones presented here (with a further tendency of

x^ ¼ x=a; t^ ¼ ðtV 0 Þ=a; g^ ¼ gðtÞ=g m ; B0 ¼ ðb0 aÞ=V 0 ; Λ ¼ ðλg m =b0 Þ: then, Eq. (9) becomes 8 ∂u^ ^ þ ∂u^ ðx^ ; tÞ ^ ¼ B0  ðx^ ; tÞ > ^ > > ∂t^ 8 ∂x 9 > > > < 0; 1Λ R x^ þ1 > ^ ^ = > ^ 1;0g ΦðsxÞ maxf x > ^ > ^ x^ ; tÞ; uð > > max: ^ ^ ^ ^ ; > vðs; tÞds > > > > > < ∂v^ ðx^ ; tÞ ^ ¼ B0  ∂t^ 8 9 > > < 0; 1Λ R x^ þ1 ^ x^ Þ = > Φð s > ^ maxf x 1;0g > max ^ ^ x^ ; tÞ; > uð > > : v^ ðs^ ; tÞd ; ^ s^ > > > > > > > ^ x^ ; 0Þ ¼ 0; v^ ðx^ ; 0Þ ¼ 0; 8 x^ 4 0; uð > > > :^ ^ ¼ gð ^ ^ tÞ; uð0; tÞ 8 t^ 40:

ð10Þ

The above Eq. (10) were numerically solved by using an up-wind scheme (Heinrich et al., 1977). In order to assess the influence of

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the non-local interaction term on the asymptotic behavior of the solutions, we ran three numerical experiments with decreasing values of Λ. Our first numerical experiment was done with parameters given by B0 ¼ 2 and Λ ¼ 2, and the continuous function ^ gðtÞ defined as one on the interval [1, 2], as zero for t Z3, and as linear functions on the intervals [0, 1] and [2, 3]. At time t^ ¼ 15, the ^ values for uand v^ were practically stabilized as predicted by the ^ theory (at that time the highest value for uwas of order 106 ). The final pattern of the buried larvae, which is given by the graph of v^ as a function of x^ , is shown in Fig. 2(a). We observe that, up to numerical error, the numerical solution v^ satisfies the theoretical R1 R1 ^ requirement 0 v^ ðxÞdx ¼ 0 gðtÞdt. In fact, in the case of Fig. 2(a), R1 R1 ^ gðtÞdt ¼ 2 and numerically it is computed that 0 v^ ðxÞdx ¼ 0

2:00 7 0:06. The error is due to the rather crude value of the spatial discretization (in this case the spatial mesh width was 0.1). To obtain better results one could decrease the spatial mesh width R1 R1 ^ and use 0 v^ ðxÞdx ¼ 0 gðtÞdt as an effective criterion of convergence. A subsequent numerical experiment was done by taking B0 ¼ 2 and Λ ¼ 0:5, which decreases the relative influence of the ^ non-local interaction. The function gðtÞ was the same as before. At the same time as above, the results were again practically stabilized and the final pattern of the buried larvae is given by the graph in Fig. 2(b). Finally, we took Λ ¼ 0:1 and the parameter ^ B0 and the function gðtÞ as before. Again at the same time the stabilization of the results occurred, and the final graph for v^ as a function of the distance x^ is shown in Fig. 2(c). The numerical experiments did establish a connection between the influence of the non-local interaction and the existence of oscillations, with a decrease in the former causing a decrease in the latter. The mathematical framework presented here thus predicts that there will be a relationship between neighborhood influence and the appearance of oscillations in dispersal (in fact, the non-dimensional parameter Λ ¼ λg m =b0 can be increased (or decreased) by relatively increasing (or decreasing) the parameter λ that measures the effectiveness of the influence of the neighborhood). There is also a threshold in that if Λ is less than a certain value, probably ^ depending on gðtÞ and B0 , the oscillations disappear altogether (the minimum value of Λ for which these oscillations appear is presently under investigation). The importance of this non-local interaction to the appearance of oscillations is reinforced by the fact that they also appear with other transport mechanisms: if we replace the advection term in (8) by a pure diffusion term ðD∂2 u=∂x2 Þ, then we have a new

v^

v^

1.2

2.5

model (already in non-dimensional form): 8 ∂u^ ^ ^ ∂2 u^ ^ ^ > > > ∂t^ ðx; tÞ ∂x^ 2 ðx; tÞ ¼ B1  > 8 9 > > R > < 0; 1Λ x^ þ1 > > Φðs^ x^ Þ = ^ > maxf x 1;0g > ^ ^ x^ ; tÞ; uð > > max: > ; ^ s^ > v^ ðs^ ; tÞd > > > > < ∂v^ ^ ^ ðx; tÞ ¼ B1  ∂t^ 8 9 R > > < 0; 1Λ x^ þ1 = > ^ ^ Φð s  x Þ > ^ maxfx1;0g > ^ > ^ x^ ; tÞ; uð > max > : v^ ðs^ ; tÞd ; > ^ s^ > > > > > > ^ x^ ; 0Þ ¼ 0; v^ ðx^ ; 0Þ ¼ 0; 8 x^ 4 0; uð > > > > : uð0; ^ ¼ gð ^ ^ ¼ 0; 8 t^ 4 0: ^ ^ tÞ; ^ tÞ uðþ1; tÞ

The new non-dimensional time is now t^ ¼ tD=a2 and B1 ¼ b0 c=a2 , the other coefficients and non-dimensional variables are the same as before. By performing a numerical experiment similar to the previous one, with B0 ¼ 2 and Λ ¼ 2, we now obtain the oscillations depicted in Fig. 3, at time t^ ¼ 15. A combination of transport mechanisms, as advection together with diffusion, leads to the following equations (in non-dimensional form): 8 ∂u^ ^ ^ ∂u^ ^ ^ ∂2 u^ ^ ^ > > ^ ðx; tÞ þ ∂x^ ðx; tÞD0 ∂x^ 2 ðx; tÞ ¼ B0  > > ∂t 8 9 > R > > < 0; 1Λ x^ þ1 > ^ ^ = > maxfx^ 1;0g ΦðsxÞ > ^ > ^ x^ ; tÞ; max uð > > : v^ ðs^ ; tÞd ; ^ s^ > > > > > > < ∂v^ ðx^ ; tÞ ^ ¼ B0  ∂t^ 8 9 R > > < 0; 1Λ x^ þ1 = ^ ^ > Φð s  x Þ > ^ maxfx1;0g > max ^ ^ x^ ; tÞ; > uð > > : v^ ðs^ ; tÞd ; ^ s^ > > > >   > > > ^ x^ ; 0Þ ¼ 0; v^ x^ ; 0 ¼ 0; 8 x^ 4 0; uð > > > > : uð0; ^ ¼ gð ^ ^ ^ tÞ; tÞ v^ ðþ1; 0Þ ¼ 0; 8 t^ 4 0:

ð12Þ

The non-dimensional variables and coefficients are defined exactly as in Eq. (10); D0 ¼ D=aV 0 . Again the oscillations appear in a numerical experiment with B0 ¼ 2, Λ ¼ 2 and D0 ¼ 1:0, as is depicted in Fig. 4, both at time t^ ¼ 15. In Eqs. (11) and (12) the oscillations tend to disappear as Λ decreases. In fact, Figs. 2(a) and (b) resemble the oscillations observed by Godoy et al. (1995) in the experimental populations of the two invading species C. putoria and C. megacephala, whereas Fig. 2(c) seems to capture the lack of waves in the dispersal of larvae in the native species C. macellaria (Godoy et al., 1995).

v^ 3.5 3

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ð11Þ

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0

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5

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0

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Fig. 2. Theoretical dispersal pattern obtained from the numerical experiments of Eq. (10) for decreasing values of the parameter Λ. v^ Is plotted as a function of x^ for B0 ¼ 2:0 and Λ ¼ 2:0 (a), Λ ¼ 0:5 (b) and Λ ¼ 0:1 (c).

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v^ 1 0.8 0.6 0.4 0.2 0

0

5

10

15

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Fig. 3. Theoretical dispersal pattern obtained from the numerical experiments of Eq. (11) for B1 ¼ 2 and Λ ¼ 2.

v^ 1 0.8 0.6 0.4 0.2 0

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10

15 x^

Fig. 4. Theoretical dispersal pattern obtained from the numerical experiments of Eq. (12) for B0 ¼ 2, Λ ¼ 2, and D0 ¼ 1.

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C. megacephala, whose values for k are 2.45 and 1.09, respectively (Godoy et al., 1996). Therefore, the native species, C. macellaria, has the highest larval aggregation and, as a corollary to our conjectures, a presumptive smaller neighborhood size and an observed monotonically decaying dispersal. On the other hand, the two invading species have; lower levels of aggregation relative to C. macellaria and, again by inference, relatively larger neighborhood sizes and an observed oscillatory decaying dispersal. The theoretical approach followed here leads to a mathematical model involving integro-differential equations. Equations of similar sort also appear in Murray (1990) primarily in the context of models of morphogenesis. However, the integral term in (1) in this paper has an entirely different interpretation from most of the models of Murray in which similar expressions represent nonlocal dynamics and may be an alternative to terms corresponding to random diffusion ðD∂2 u=∂x2 Þ. In our case, the larvae are responding to the presence of buried larvae in a neighborhood, introducing a non-local interaction term in the model. This concept of non-local effects proved useful here to explain the appearance of waves in the horizontal distribution of larvae of blowflies, as a result of the interaction between dispersing larvae and the burrowing larvae. Obviously, the details of the final distribution of buried larvae (like the number of oscillations, their amplitude, and so on) depend on the actual type of transport mechanisms and the kind of kernel used in the non-local interaction, and there is much room for investigation on these subjects. However, the appearance of oscillations does not depend on these details, as our numerical experiments show, and these results underscore the importance of non-local effects. The overall potential for the application of integral equations in population ecology was recently addressed by Kot (1992), who obtained a variety of traveling wave solutions for problems in population growth coupled with dispersal and predator–prey systems, and also stressed the importance of the kernel formulation in this approach (Kot, 1992, p. 305). We believe that the approach advanced by Murray (1990) and Kot (1992) will find a wide range of applications in problems in population and evolutionary biology.

3. Conclusion

Acknowledgments

From a biological standpoint, the mathematical results suggest that neighborhood influence, acting as the mechanism for the inhibition of pupation, plays a crucial role in determining the form of horizontal frequency distribution of pupae in the native and introduced blowfly species. Apparently, the neighborhood influence of the native species, C. macellaria, is relatively weaker and does not produce waves in the distribution of dispersing larvae, whereas the reverse occurs for the introduced species, C. megacephala and C. putoria. Neighborhood influence, in particular its size a and the parameter λ, could be estimated in experiments of dispersal using measures of aggregation, under the expectation that aggregation will be antagonistically affected by the strength of inhibition of pupation. In other words, the stronger the effect of inhibition on pupation the less pronounced the aggregation of larvae should be, and therefore the higher the likelihood of oscillations in the dispersal pattern. We (Godoy et al., 1996) have estimates for aggregation during dispersal for the three blowfly species based on the parameter k in the negative binomial distribution, which is commonly used as a measure of aggregation (Ludwig and Reynolds, 1988). In this statistics, the larger the value of k the lowest the level of aggregation (Ludwig and Reynolds, 1988). The native species, C. macellaria, has the lowest value (0.39) for the parameter k, compared with the invading species, C. putoria and

We thank the anonymous reviewers for useful comments and suggestions that helped us to improve the manuscript. We are particularly indebted to one of the reviewers for suggesting the inclusion of alternate mechanisms of larval movement that highlighted the importance of non-local effects for immature dispersal, and also for pointing out a simple numerical criterion of convergence for the integrals describing the density of larvae buried in the substrate. This work was funded through grants from Fundação de Amparo à Pesquisa do Estado de São Paulo (Nos. 83/2740-8, 83/4244-8, and 84/7920-1). Research by the authors is partially supported by the Conselho Nacional de Desenvolvimento Científico e Tecnológico. References Bengtsson, G., Hedlund, K., Rundgren, S., 1994. Food- and density-dependent dispersal] evidence from a soil collembolan. Journal of Animal Ecology 63, 513–520. Boldrini, J.L., Moretti, A.C., 1995. Patterns of post-feeding larval dispersion in blowflies. Technical Report UNICAMP-IMECC 24, 1–21. de Jong, G., 1979. The influence of the distribution of juveniles over patches of food on the dynamics of a population. Netherlands Journal of Zoology 29, 33–51. Felsenstein, J., 1994. Theoretical Evolutionary Genetics. University of Washington Press, Seattle. Gaines, S.D., Bertness, M., 1993. The dynamics of juvenile dispersal: why field ecologists must integrate. Ecology 74, 2430–2435.

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J.L. Boldrini et al. / Journal of Theoretical Biology 336 (2013) 252–258

Gilpin, M., 1991. The genetic effective size of a metapopulation. Biological Journal of the Linnean Society 42, 165–175. Ginzburg, L.R., 1983. Theory of Natural Selection and Population Growth. The Benjamin: Cummings Publishing Company, Menlo Park. Godoy, W.A.C., Reis, S.F., Von Zuben, C.J., Ribeiro, O.B., 1993. Population dynamics of Chrysomya putoria (Dipt., Calliphoridae). Journal of Applied Entomology 116, 163–169. Godoy, W.A.C., Fowler, H.G., Von Zuben, C.J., Ziti, L., Ribeiro, O.B., 1995. Larval dispersion in Chrysomya megacephala, Chrysomya putoria and Cochliomyia macellaria (Dipt., Calliphoridae). Journal of Applied Entomology 119, 263–266. Godoy, W.A.C.G., Von Zuben, C.J., Reis, S.F., 1996. Larval dispersal in Chrysomya megacephala, Chrysomya putoria and Cochliomyia macellaria (Dipt., Calliphoridae): ecological consequences of aggregation behavior. Journal of Applied Entomology 120, 423–426. Greenberg, B., 1990. Behavior of postfeeding larvae of some Calliphoridae and a Muscidae (Diptera). Annals of the Entomological Society 83, 1210–1214. Guimarães, J.H., Prado, A.P., Linhares, A.X., 1978. Three newly introduced blowfly species in Southern Brazil (Diptera: Calliphoridae). Revista Brasileira de Entomologia 22, 53–60. Hanski, I., 1987. Carrion fly community dynamics: patchiness, seasonality and coexistence. Ecological Entomology 12, 257–266. Heinrich, J.C., Huyakorn, P.S., Mitchell, A.R., Zienkiewicz, O., 1977. An “upwind” finite element scheme for two-dimensional convective transport equation. International Journal for Numerical Methods in Engineering 11, 131–143. Ives, A., 1989. The optimal clutch size of insects when many females oviposit per patch. American Naturalist 133, 671–687. Ives, A., 1991. Aggregation and coexistence in a carrion fly community. Ecological Monographs 61, 75–94. Kot, M., 1992. Discrete-time travelling waves: ecological examples. Journal of Mathematical Biology 30, 413–436. Kotaki, T., Fujii, H., 1995. Crowding inhibits pupation in Tribolium freemani: contact chemical and mechanical stimuli are involved. Entomologia Experimentalis et Applicata 74, 145–149. Levot, G.W., Brown, K.R., Shipp, E., 1979. Larval growth of some calliphorid and sarcophagid Diptera. Bulletin of Entomological Research 69, 469–475.

Ludwig, J.A., Reynolds, J.F., 1988. Statistical Ecology: A Primer on Methods and Computing. John Wiley and Sons, New York. Mueller, L.D., 1986. Density-dependent rates of population growth: estimation in laboratory populations. American Naturalist 128, 282–293. Mueller, L.D., 1988. Density-dependent population growth and natural selection in food-limited environments: the Drosophila model. American Naturalist 132, 786–809. Murray, J.D., 1990. Mathematical Biology. Springer-Verlag, Berlin. Nagylaki, T., 1992. Introduction to Theoretical Population Genetics. Springer-Verlag, Berlin. Nakakita, H., 1982. Effect of larval density on pupation of Tribolium freemani Hinton (Coleoptera: Tenebrionidae). Applied Entomology and Zoology 17, 269–276. Okubo, A., 1980. Diffusion and Ecological Problems: Mathematical Models. Springer Verlag, Berlin. Prout, T., Mcchesney, F., 1985. Competition among immatures affects their adult fertility: population dynamics. American Naturalist 126, 521–558. Rohani, P., Miramontes, O., 1995. Immigration and the persistence of chaos in population models. Journal of the Theoretical Biology 175, 203–206. Roughgarden, J., Gaines, S.D., Possingham, H., 1988. Recruitment dynamics in complex life cycles. Science 241, 1460–1466. Taylor, R.A.J., 1977. The relationship between density and distance of dispersing insects. Ecological Entomology 3, 63–70. Taylor, L.R., Taylor, R.A.J., 1977. Aggregation, migration and population mechanics. Nature 265, 415–421. Tschinkel, W.R., Willson, C.P., 1971. Inhibition of pupation due to crowding in some tenebrionid beetles. Journal of Experimental Zoology 176, 137–146. Von Zuben, C.J., Reis, S.F., Val, J.B.R., Godoy, W.A.C., Ribeiro, O.B., 1993. Dynamics of a mathematical model of Chrysomya megacephala. Journal of Medical Entomology 30, 443–448. Von Zuben, C.J., Bassanezi, R.C., Reis, S.F., Godoy, W.A.C., Von Zuben, F.J., 1996. Theoretical approaches to forensic entomology. I. Mathematical model of postfeeding larval dispersal. Journal of Applied Entomology 120, 379–382. Yto, Y., 1977. Contact stimulation between intact and amputated deuteronymphs of Parasitus gregarious for induction of molting. Applied Entomology and Zoology 12, 290–291.