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Seasonal dispersal of pests: one surge or two?

M. D. PAULSON,* A. I. HOUSTON,? J. M. MCNAMARA* & R. J. H. PAYNE?

*Department of Mathematics, University of Bristol, Bristol, UK

?School of Biological Sciences, University of Bristol, Bristol, UK

Introduction

Evolutionary theory has an important role in predicting

how species will respond to climatic change. Dispersal

theory, in particular, will be pertinent to understanding

how the ecological dynamics of agricultural pests might

changeastheenvironmentchanges.Previousstudieshave

looked ata wide range of explanationsfor theevolution of

dispersal (Johnson & Gaines, 1990; Bowler & Benton,

2005; Ronce, 2007), such as avoidance of competition for

resources (Travis et al., 1999; Poethke & Hovestadt, 2002),

reduction in inbreeding (Bengtsson, 1978; Motro, 1991),

avoidance of kin competition (Hamilton & May, 1977;

Comins, 1982; Plantegenest & Kindlmann, 1999; Kisdi,

2004) and as a bet-hedging strategy (Philippi & Seger,

1989; Krug, 2001). In this paper, we focus on the role of

seasonalityinshapingdispersalbehaviour,bothbecauseit

has been somewhat neglected in previous theory and it is

likely to be highly relevant to understanding the dispersal

behaviour of a wide range of major agricultural pests.

Time of season as a cue for dispersal has been studied

in relatively few models. Cohen (1967) modelled the

optimal fraction of a population that should migrate to

avoid an unfavourable period of the season. There is a

trade-off between the risk of migrating to the wintering

territory and the chances of nonmigratory individuals

surviving in the breeding territory. The model predicted

that the proportion of individuals that migrate would be

strongly influenced by factors affecting the nonmigratory

individuals, such as crowding and food availability.

Macevicz & Oster (1976) studied the effect of season on

the optimal reproductive strategy for eusocial insects.

Their model predicted that a colony should split repro-

ductive effort into two distinct phases during the season.

Although neither of these papers considered a context

directly comparable with the one we are interested in,

they both contain key insights. Cohen’s model under-

scores the importance of local conditions on dispersal

behaviour; the model of Macevicz and Oster highlights

the importance of timing in a seasonal context.

We are interested in the dispersal behaviour of species

with several generations during each season, as is the

case for many pest species. One might describe such a life

history as involving a ‘seasonal metapopulation’: after

overwintering there is a period of reproduction in

separate colonies with occasional dispersal to found

new colonies, typically followed by a separate sexual

phase of dispersal, mating and egg laying. In some

Correspondence: Robert J. H. Payne, School of Biological Sciences,

University of Bristol, Bristol BS8 1UG, UK.

Tel: +44 117 928 8254; fax: +44 117 331 7985;

e-mail: robert.payne@bristol.ac.uk

ª 2009 THE AUTHORS. J. EVOL. BIOL. 22 (2009) 1193–1202

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Keywords:

agricultural pest;

aphid;

climate change;

dispersal;

dynamic programming;

evolutionarily stable strategy;

metapopulation;

season.

Abstract

Many agricultural pest species occur in seasonal metapopulations with a

period of asexual reproduction. We use evolutionary theory to predict timing

of dispersal for such species, and identify four sequential phases: no dispersal,

dispersal from initially occupied patches, dispersal from later colonized

patches, and no dispersal. The third type of phase occurs only when

reproductive rates are relatively high; we speculate that this could explain

why among aphids there can be either one or two waves of dispersal during a

season, depending on the species. Our model also explains other features of

aphid biology, including a summer crash in colony size, and a decline in the

number of colonies towards the end of each reproductive season. The presence

of an additional surge of dispersal becomes more likely as season length

increases, and does not require further evolution. This could have profound

implications for pest management during future climatic warming.

doi:10.1111/j.1420-9101.2009.01730.x

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species, there is a distinct asexual dispersal morph; this

would seem to underline the evolutionary importance of

the asexual dispersal behaviour. In the following, we

only consider the asexual dispersal phase.

Aphids are a classic example of species which have a

seasonal metapopulation, and which are important agri-

cultural pests: seasonality is known to be a key compo-

nent of aphid population dynamics (Sequeira & Dixon,

1997), and we shall use various examples from aphid

population biology to illustrate some of our arguments.

Nonetheless, we keep our model generic and do not

explicitly model a separate dispersal morph, as we wish

to focus on the underlying mechanisms that ultimately

drive the timing and contingency of dispersal of such

species, rather than dwelling on proximate details.

Many features of plant and animal life histories

correlate with seasonal changes in temperature, light

levels and/or day length. In the case of aphids, asexual

females hatch from diapausing eggs in early spring and

then reproduce and disperse at varying rates until

autumn. The shortening days and lower temperatures

induce the production of sexual males and females who

mate to produce eggs, ready to hatch in the following

spring when temperatures increase. Various studies

have shown aphid dispersal to be mediated by time of

season (Dixon, 1998). For example, dispersal rarely

occurs at the end of the season (Massonnet et al., 2002;

Weisser & Ha ¨rri, 2005), and the hop aphid, Phorodon

humuli, has a peak of dispersal during spring, followed

by a lull in summer then another peak in autumn

(Dixon, 1998).

We use dynamic programming (Bellman, 1957) to find

the evolutionarily stable strategy (ESS) (Maynard Smith,

1982), for dispersal behaviour in a seasonal metapopu-

lation. It is known that aphids can detect the local density

of individuals on a patch (Sutherland, 1969), food quality

(Dixon & Glen, 1971) and time of season (Matsuka &

Mittler, 1978), either directly or via a proxy. It would

seem likely that these three types of information can be

detected by most agricultural pests. We therefore assume

that this information is known to individuals at the time

of making a dispersal decision, and so we seek the ESS

dispersal behaviour as a function of density, patch quality

and time of season.

Methods

Our basic model represents a single-species metapopula-

tion in an environment consisting of many habitat

patches linked by dispersal. Each patch can either hold

a colony or be empty. All dispersers are assumed to be

equally likely to arrive at any patch, and so spatial

structure is treated only implicitly. All individuals are

parthenogenetic, can disperse and possess the same

intrinsic properties. Each patch supports n individuals

up to the patch capacity (or patch ‘quality’), k, which

may change over time.

The length of a season is T weeks. The per capita

dispersal rate is A(n,k,t), so during each time step, Dt,

each individual on a patch disperses with probability

A(n,k,t)Dt. For simplicity, we assume growth follows a

logistic equation, with per capita intrinsic reproductive

rate k, and patch capacity k. Given the number of

individuals n on a patch at time t, the number of

individuals n0on a patch at time t + Dt is therefore:

n0¼ n þ n k 1 ?n

This equation yields a real number n0that is then mapped

onto a grid of integers as described in Houston &

McNamara (1999). At the start of the season a small

proportion of patches p0 (founder colonies) are each

occupied by n0individuals, with all other patches initially

empty.

In our model, a dispersing individual has probability q

of successfully arriving at a patch. Dispersers settle on

patches at random. It is assumed that settling on an

already occupied patch is a wasted journey (such as

observed in aphids; see Gange, 1985) and as a first

approximation these individuals die. Similarly, we

assume that if during one time step more than one

individual arrives at an empty patch, one individual

chosen at random colonizes the patch and the rest die.

Consumption of patches can occur at a rate d(n,k) and

seasonal growth and decay of patches at a rate g(t). Given

patch capacity k at time t, the patch capacity k0at time

t + Dt is:

k0¼ k þ ðgðtÞ ? dðn;kÞÞDt

In a later section, we consider the effect of patch

dynamics by allowing d(n,k) and g(t) to be nonzero, but

for now we use the simplest scenario, in which patch

capacity is constant, i.e. k0¼ k. Under this scenario, all

patchesintheenvironmentstartataninitialcapacityBand

do not vary throughout the season. This is a state-based

model (Houston & McNamara, 1999; Clark & Mangel,

2000)inwhichthestateunderconsiderationisthestateof

a colony, in terms of its density and patch capacity.

To find the ESS, competition is implemented in the

form of a wild-type and mutant genotype. We assume

that each patch can only be occupied by one genotype at

a time. Genotypes are identical apart from their dispersal

strategy A(n,k,t). We simulate a wild-type strategy over a

season to find the distribution of wild-type across all

patches at each time step. The simulation outputs ak(t),

the probability that a wild-type migrant successfully

colonizes a patch with capacity k at time t. It is assumed

that a mutant migrant will experience the same condi-

tions as a wild-type migrant because mutants are rare

and so will only have negligible impact on the environ-

ment. We can therefore pass ak(t) to the dynamic

programming stage (see below) to be used as the

probability of a mutant migrant colonizing a patch with

capacity k at time t. For more details on the wild-type

simulation see Appendix.

k

??

? Aðn;k;tÞ

??

Dt

ð1Þ

ð2Þ

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The reproductive success of a mutant strategy is taken

to be the expected number of descendants left at the end

of the season by n0mutants present on a patch at the

beginning of the season, and this is used as the measure

of fitness for calculating the ESS. We let F(n,k,t) denote

the expected number of mutants, produced from time t

onwards, alive at the end of the season that are directly

descended from n mutants currently on a patch with

capacity k, where k belongs to a set of possible states ks.

Then, for a given set of state variables n and k at time t,

the dynamic programming equation for finding the form

of A(n,k,t) that maximizes F(n,k,t) is:

n

þ nAðn;k;tÞDt

Fðn;k;tÞ ¼ max

Aðn;k;tÞ

Fðn0;k0;t þ DtÞ

X

i2ks

aiðtÞFð1;i;t þ DtÞ

o

ð3Þ

Dynamic programming is used to work backwards from

the end of the season, finding the optimal value of

A(n,k,t) for all values of n, k and t. The measure of success

for a mutant dispersal strategy on a patch with state

variables n and k at the end of the season (t ¼ T) is taken

to be the total number of individuals alive at the end of

the season, i.e. F(n,k,T) ¼ n. Once the optimal mutant

strategy has been computed, this is then set as the new

wild-type strategy in the next computation. When the

model is run and the mutant and wild-type strategies

converge we obtain a Nash equilibrium (Nash, 1951).

Details on the computational procedure and calculation

of the wild-type are in the Appendix.

Results

Evolutionarily stable dispersal strategies were readily

computed for most choices of parameters. Figure 1a and

b shows contour plots of the evolutionarily stable

dispersal strategy as a function of all possible values of

time of season and local density (patch capacity is

0

10

6000

20

30

40

50

60

70

80

90

100

0510 15 20

05

Time of season (weeks)

10 1520

Colony size

Total number of dispersers

Time of season (weeks)

0.8

0.6

0.4

0.2

5000

4000

3000

2000

1000

0

Phase A

Phase B

Phase DPhase D

(a)

0

10

20

30

40

50

60

70

80

90

100

05 10 1520

Colony size

Time of season (weeks)

0.8

0.6

0.4

0.2

(b)

(c)

05

Time of season (weeks)

1015 20

6000

Total number of dispersers

5000

4000

3000

2000

1000

0

Phase A

Phase B

Phase C

(d)

D

Fig. 1 Evolutionarily stable strategy (ESS) dispersal solution for the basic model. (a) Contour plot of per capita dispersal rate, A(n,k,t),

dependent on time and density, for the ESS solution with per capita intrinsic reproductive rate k ¼ 0.7 week)1. The plot includes two example

trajectories, illustrating the number of individuals on a founder colony (solid line) and a secondary colony (dashed line) during the season.

(b) Same, but with k ¼ 1.2 week)1. (c) Total number of dispersers from founder (solid line) and other (dashed line) colonies during a season

when the ESS with k ¼ 0.7 week)1is followed (assuming 1000 patches). (d) Same, but with k ¼ 1.2 week)1. Other parameters: k ¼ 100,

T ¼ 20 weeks, p0¼ 0.05, n0¼ 1 and q ¼ 0.05.

Seasonal dispersal of pests

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constant throughout the season), for two different values

of the per capita intrinsic reproductive rate k. Whilst this

is the strategy that is followed by each individual, it is

useful to examine the expected behaviour of the system

as a whole when this strategy is followed. How the state

of a particular patch changes over time can be repre-

sented as a trajectory that traverses the state space. The

thick solid lines in Fig. 1a and b show such a trajectory

for one of the original founder colonies. The thick dashed

lines show a trajectory for one of the colonies founded

during the season (secondary colonies).

Four seasonal phases

Figure 1c and d shows the number of dispersers in the

population, decomposed into those from founder and

later colonized patches, for a typical realisation of the

computations following the ESS in Fig. 1a and b respec-

tively. We identify the four phases of behaviour as

follows:

Phase A: No dispersal at start of season. The number of

individuals on patches is low; so, there is little local

competition for resources, and no need to disperse.

Phase B: Primary dispersal driven by local competition on

founder patches. As the population increases the local

growth rate diminishes. A newly founded colony thus

has a growth rate that is greater than that of the

original colonies, and over time the relative benefit of

founding a new colony increases. Phase B starts when

the long-term benefits of founding a new colony start

to outweigh the short-term risks of dispersal; so, a

moderate rate of dispersal becomes favourable.

Phase C: Additional dispersal driven by local competition

on patches colonized during phase B. That is to say,

phase C type dispersal is attributed to the descendants

of participants of phase B type dispersal. This period of

increased dispersal will occur only if the season is

sufficiently long. It is possible that subsequent phases

of dispersal could occur again for the same reason,

although this was not observed for the parameter

values used in our simulations.

Phase D: No dispersal towards end of season. To benefit

from dispersing to found a new colony there needs to

be sufficient time for that colony to grow. As the end of

the season approaches the time available diminishes.

Phase D starts when the diminishing benefits of a new

colony become less than the costs of dispersal. Note

that the timing of this phase is dependent on the time

before the end of season, rather than the time from the

start of season.

Global competition for patches does not play a role in

determining the presence of phases B or C. We confirmed

this by varying the initial proportion of patches occupied,

p0, and noting that the phases did not alter. We also

varied the initial number of individuals on each founder

patch, n0. We found that as n0is increased: (i) phase B

starts earlier, (ii) three-phase behaviour (phase types A,

B and D) changes to four-phase behaviour (phase types

A, B, C and D) and (iii) phase C, if present, becomes

longer. This evidence, along with the results in Fig. 1c

and d, supports the premise that phase C is driven by

local competition on patches that were colonized in

phase B. In the next section, we examine under what

conditions phase C may or may not appear.

Two peaks of dispersal

During phase B, dispersal is driven by competition for

local resources on founder patches. Local competition

also drives phase C type behaviour, but this depends on

the rate at which the patches colonized during phase B

become crowded. This can be tested by examining how

the ESS solution changes depending on the per capita

reproductive rate, k. Here, we assume that k is constant

across the season; we do not consider the case where k

changes over a season, but see below for a discussion of

what happens when patch capacity varies over a

season.

When k is small each colony grows more slowly; so,

phase B is expected to start later and patches colonized

during phase B will fill up less quickly, meaning local

competition on these patches does not become significant

before phase D starts, and thus there should be less

chance of phase C occurring. Our computations support

this explanation: we find that phase C only appears when

k is sufficiently large. This is illustrated in Fig. 2a, which

shows the total number of dispersers for two different

values of k. When k is small, numbers on colonized

patches never reach a high enough level to induce phase

C type dispersal (also phase B type dispersal occurs later

and more briefly).

Figure 2b plots the same results as Fig. 2a, but instead

of showing the total number of dispersers in the popu-

lation it shows the average dispersers per occupied patch.

This allows us to examine the behaviour at a colony

level. Our results indicate that, at the colony level, two

peaks (or ‘waves’) of dispersal are observed for higher

values of k.

Peaks of aphid dispersal during a season have been

observed by Dixon (1998), although the number of peaks

depended on the species. In some species there were two

peaks, with the first peak in spring and the second in

autumn. In other species, Dixon found only a single peak

of dispersal during a season. According to Dixon, the

species of aphid with more than one dispersal peak

during the season are host alternating, with one of the

dispersal peaks due to dispersal from primary to second-

ary host plants in the spring, and another due to dispersal

from secondary to primary host plants in the autumn.

Dixon suggested that dispersing between host plants that

differ in their seasonal growth phases could allow aphids

to better exploit a continuous supply of high-quality

food. Our model does not allow for more than one type

of ‘host’ patch, but one might speculate that if a species

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starts to experience phase C type dispersal there would be

strong selective pressure to switch to colonization of a

secondary host type. Our model predicts that species with

the higher reproductive rate are most likely to develop

two peaks of dispersal during a season, and one might

argue that these are the most likely to evolve host

switching behaviour.

The rate at which patches become crowded is not only

affected by the basic reproductive rate, but also by the

overall season duration, T. We predicted that phase C

type dispersal is more likely when the season is long. This

is supported by our computations. Figure 3 shows the

effect of varying the length of season T on dispersal

during a season. It can be seen that large T has a similar

effect to large k: a single dispersal peak during a season

changes to two peaks of dispersal as T increases. This

result suggests that a change in the season length

experienced by a species could lead to a qualitative

change in dispersal behaviour.

Environmental change

We have just described how having a longer season or a

higher growth rate can lead to two peaks of dispersal in a

season, provided the species is fully adapted to its

environment. However, in the context of climatic

change, we must ask what would happen if the season

length, or growth rate, has increased yet there has not

been sufficient time for the species to adapt to the new

environment.

Suppose, for example, that climatic change leads to

higher ambient temperatures which increase the repro-

ductive rate k. We examined this scenario by finding the

ESS under k ¼ 0.75 (for which only one peak of dispersal

is manifest), and then observing what would happen if

that strategy were used, without further adaptation,

under k ¼ 1.2 (cf. discussion in Fero ´ et al., 2008). We

found that the ESS adapted to k ¼ 0.75 showed two

peaks of dispersal under the k ¼ 1.2 environment. In

0

500

1000

1500

2000

2500

3000

3500

4000

0

5 10

15

20

Total number of dispersers

Time of season (weeks)

0

2

4

6

8

10

12

14

16

18

05

10

15 20

Mean dispersers per colony

Time of season (weeks)

Low

High

λ

0

5

10

15

20

0 0.5

1 1.5

Time of season (weeks)

4

8

12

16 20

(a)

(b)

(c)

λ

High λ

High λ

λ

Low λ

Low λ

Fig. 2 Effect of varying the per capita reproductive rate, k. (a) The

total number of dispersers. (b) The number of dispersers per colony.

(c) Contour plot showing how the number of dispersers per colony

can have one or two peaks during a season depending on the

reproductive rate. The values of k used for the two cross-sections

shown in (a) and (b) are marked by two vertical lines. For all panels,

dashed line: k ¼ 0.6 week)1; solid line: k ¼ 1.0 week)1. Other

parameters as in Fig. 1.

0

2

4

6

8

10

12

14

16

18

05 1015 20

Mean dispersers per colony

Time of season (weeks)

Low T

High T

Fig. 3 Effect of varying the length of the season, T. Dashed line: T ¼

12 weeks (season starts at t ¼ 4 and finishes at t ¼ 16); solid line:

T ¼ 20 weeks (season starts at t ¼ 0 and finishes at t ¼ 20). Other

parameters as in Fig. 1, but with k ¼ 1 week)1.

Seasonal dispersal of pests

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