Host-parasitoid models in temporally and spatially varying environment

ELSEVIER Ecological Modelling 75/76 (1994) 161-170

£(OI, OOEIII, IIIOIMWII6

Host-parasitoid models in temporally and spatially varying environment

Thorsten Wiegand *, Christian Wissel

Umweltforschungszentrum Leipzig-Halle, Permoserstr. 15, 04318 Leipzig, Germany

Abstract

So far models describe host-parasitoid interaction in a phenomenological way. We discovered that the parasitoid behaviour of searching and laying eggs is essential for the question of stability. Therefore, we model the searching of parasitoids in a spatial and temporal inhomogeneous environment. Temporal inhomogeneity results in seasonal patterns of the distribution of hosts and parasitoids. We find that a slight temporal desynchronization between the emergence of hosts and parasitoids is an effective mechanism leading to stability. In the case of inhomogeneous space we solve some as yet unresolved problems. The existing host-parasitoid models are confined to very special cases because they use a constant spatial distribution of parasitoids and hosts from generation to generation. We show that variation in this distribution can appear and is important for coexistence.

Key words: Host-parasite interaction; Spatial distribution

I. Introduct ion

Intensive studies have been done to investigate arthropod predator-prey or host-parasitoid systems (Hassell, 1978; Waage and Greathead, 1986). Many key- factors which lead to stability or coexistence could be identified. Varley and Gradwell (1963), Beddington et al. (1975) and Southwood and Comins (1976) worked out that density-dependent regulation of the host population leads to stability. The same effect emerges from a spatial inhomogeneous environment (Huffaker, 1958; Huffaker et al., 1963), refuges in space (Maynard Smith, 1974; Hassell, 1978) or from aggregation of the parasitoid population in space (Hassell

* Corresponding author.

0304-3800/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0304-3800(93)E0125-M

162 T. Wiegand, C. Wissel / Ecological Modelling 75/76 (1994) 161-170

and May, 1973, 1974). All these models have been very useful for discovering key-factors on a more general level. But they consider the searching behaviour, which in our studies turned out to be of great importance, only in a phenomenological way.

We describe the searching of parasitoids for hosts and their laying of eggs in a spatially and temporally inhomogeneous environment in a more detailed and realistic manner.

The development of individuals in host and parasitoid populations is closely related to the seasons of the year. Each stage (mating, searching for hosts, laying of eggs, development from egg to adult) appears within a certain time of the year. The sequence of non-overlapping generations obviously has to be described on two different levels (separation of timescales): the seasonal level and the level of the population dynamics. On the seasonal level some important processes are: - mating, searching for hosts, laying of eggs of parasitoids; - hosts being encountered and killed. On the level of population dynamics, population size changes from generation to generation. Because of the simple interaction between host and parasitoid and the separation of time-scales, the following model describing the population dynamics of host-parasitoid systems becomes simple. We are able to include complicated spatial and temporal aspects of searching behaviour on the seasonal level without affecting the construction of the model on the level of population dynamics. Nevertheless the model remains manageable. Let R t be the number of hosts in generation t, and Pt the number of female parasitoids in generation t. Only unattacked hosts are able to reproduce. If f is the probability of one host not being attacked and L is the main reproduction rate of hosts we find in the next (t + 1) generation

Rt+ 1 = L R , f . (1)

From each encountered host m female parasitoids develop in the next generation. We get

Pt+l = mRt(1 - f ) . (2)

By rescaling Pt the full model for population dynamics can be written as:

R,+ 1 = L R t f , (3)

P,+l =Rt(1 - f ) - (4)

Here we like to emphasize that the probability f links the level of population dynamics with the seasonal level, f adds up all biological information emerging from the seasonal level about the spatial distribution of the host population, about the seasonal pattern of host and parasitoid population and about the searching strategy of parasitoids.

With this background we can formulate the aim of our model: We like to investigate the influence which arises from temporal and spatial aspects of the searching behaviour on the population dynamics. We consider certain biological factors acting on the seasonal level and determine the reaction on the population

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dynamics. This means that t he consequences of seasonal patterns of host and parasitoid populations, of spatial inhomogeneity and of specific searching behaviour of parasitoids on the coexistence and stability of both populations are investigated. By concentrating on these specific questions we ignore details which may be realistic but which are not essential for these questions. Modelling the factors of the seasonal level in a selfconsistent manner we search for general structures and tendencies. We enter this problem in the most simple way because models become more complicated by themselves.

The most simple model to start with is the Nicholson-Bailey model. The assumption of homogeneous space and time leads to the well known term:

f = e -cP' . (5)

The parameter c is the parasitoid searching efficiency. The Nicholson-Bailey model shows instable oscillations. We use this model as a reference model for our approach. Releasing its requirement of homogeneous time and space step by step we can identify the mechanisms leading to coexistence and stability.

2. Temporally varying environment

Despite a few comments in literature (Griffiths and Holling, 1969; Hassell, 1969, 1986) which suspect that desynchronization between host and parasitoid population has a stabilizing effect on population dynamics, there is no further theoretical evidence for this hypothesis. We investigate this hypothesis by extending the Nicholson-Bailey model to a model with temporally varying environment.

2.1. The model

The Nicholson-Bailey model assumes random search of parasitoids for hosts. This means that at every time during the searching period the probability of any host being found by any parasitoid is the same. Now we modify this oversimplified assumption and introduce more realistic probabilities of hosts being found. Biolog- ical mechanisms which can lead to inhomogenity in time are: the hatching of insects is triggered by various internal and external factors. The times of development of different individuals within a population differ slightly. Desynchronized life histories of individuals within a population occur. The time which parasitoids spend on search for hosts and on laying eggs differs from individual to individual. Because of these differences we obtain a temporal distribution of the number of searching parasitoids. Parasitoids are usually specialized and closely related to their host. They attack a certain life stage. Thus a host is available for parasitoids only for a limited period of time. Because of desynchronized life histories of hosts we obtain a temporal distribution of the number of available hosts.

To transform this verbal knowledge into a mathematical form we introduce new variables, fn t is defined to be the mean time which a host is suitable for parasitoids and analogously pnt is the mean time which a parasitoid is able to lay

1 6 4 T. Wiegand, C. Wissel / Ecological Modelling 75 / 76 (1994) 161-170

r(t),Y(t)

0.03

0.02

0.01

0.00 I 30 60

t

90

Fig. 1. Typical time pattern of available hosts (solid) and parastoids able to lay eggs (dashed) as used in our model.

eggs. We use t as the running time during the season, r(t)dt is the probability that a host becomes suitable for parasitoids within the interval (t,t + dt). Y(t)dt is the probability that a parasitoid becomes able to lay eggs within the interval (t,t + dt). Therefore the number R(t) of available hosts at time t can be calculated as:

t r

R(t )=Rt f t_ fn tr ( t ) d t ' where f yearr( t )d t=l (6)

and R t is the total number of hosts in this year. In the same way we calculate the number P(t) of parasitoids which are able to lay eggs:

P(t ) =Pt f t fp tY( t dt' with ~yfearY(t) d t = 1. (7)

Pt is the total number of parasitoids in year t. Fig. 1 shows typical time patterns r(t) and Y(t).

If we assume random encounters between parasitoids and hosts the probability wdt of one available host being encountered is the same for all available hosts within (t,t + dt). We obtain for a short interval dt:

w dt = c dtP(t ) , (8)

where cdt is the fraction of the host habitat which is searched through by one parasitoid in the span dt. The probability W 0 of not being encountered by P(t) searching parasitoids for one host which became available for parasitoids at the time t - f n t is:

Wo( t' + dt ' ) = Wo( t')(1 - w dt ' ) . (9)

This leads to the differential equation:

dWo(t' ) dt' wW°( t') (10)

with the solution:

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Finally the probability of not being found for any host turns out to be

f= fyearr(t - fnt)Wo(t )d t . (12)

This formula allows the calculation of probabilities f for all time patterns r(t) and Y(t). We consider specific biological situations by means of specific time patterns r(t) and Y(t) to investigate their impact on population dynamics and coexistence.

2.2. Resul~

We analyse this model by the method of local stability analysis and numerical integration of the model equations. The local stability analysis gives a locally stable equilibrium P* determined by

L fyearr(t - fn t ) e -A(t)e* dt = 1 (13)

if the following condition is fulfilled:

with

L - 1 LfeaA( t )P*r ( t - f n t ) J y . e -*(')e* dt < T (14)

1 t -~tcf_fntP(t) dt- A(t). (15)

A(t)P, measures the mean number of parasitoid attacks for hosts which are available for parasitoids within the interval (t -fnt, t). After extensive variations of the pattern r(t) and A(t) we found out that the stability behaviour of our model follows a simple rule (Wiegand, 1992): we obtain locally stable equilibria P* if a few hosts remain unattacked. This is guaranteed by a slight desynchronization of the seasonal patterns r(t) and Y(t). This rule turned out to be independent from other details of the the patterns. While slight desynchronization leads to coexistence and stability, perfect synchronization causes instability. Bad synchronization tends to produce very large equilibrium populations.

3. Spatially heterogeneous environment

In this section we introduce spatial inhomogeneous distributions of hosts and non-random search of parasitoids into our model. Inhomogeneous distribution of hosts and fragmented habitats are well described with the patch concept (Wiens, 1976; Hassell and Southwood, 1978). We follow an approach which was given first by Hassell and May (1973). They consider a habitat subdivided into n single patches and assume each patch to be homogeneous, a i is the fraction of hosts in

166 T. Wiegand, C. Wissel / Ecological Modelling 75/76 (1994) 161-170

patch i, whereas f l i is the corresponding mean fraction of parasitoids. To obtain the probability f one simply has to sum up all patches. Because of homogeneous space and time within the patches one gets (see Eq. 5):

n

f = Y'~a, e -~*cP,. (16) i = 1

This formula is - at a quick glance - well matched to the problem of inhomogenous space, but it hides deeper problems. Even if we know the host distribution a i over the patches how should we estimate the parasitoid response, the /3i? There have been several attempts to specify a~ and /3~ (May and Hassell, 1973; Hassell, 1978). But to obtain analytically manageable models, rough simplifi- cations are necessary. All these models are suffering from their exclusively spatial point of view. But host searching is a dynamic process in space and time. The combination of spatial and temporal aspects in a submodel delivers a surprisingly simple connection between a~ and /3 r

3.1. Connection between ol i and [~i

We model searching of parasitoid within a patchy environment in a detailed way. To do this we divide the searching process in two distinct parts, the searching for patches within the fragmented habitat and the searching for hosts within one patch. Then two important questions arise: 1. how do the parasitoids find patches? 2. how long does a parasitoid stay in one patch?

Both questions are empirically well investigated (Hassell, 1978; Lessels, 1985; Van Alphen and Vet, 1986). To include all this knowledge into a submodel we find two decisive variables which are closely related to the two questions above: gi(ai) is the probability that patch i is choosen if a parasitoid finds a patch. It measures the relative attractivity of patch i. Zi(oli) is defined to be the time which a parasitoid stays in patch i. The searching parasitoids repeat the following be- havioural pattern: 1. the parasitoid searches for patches and eventually encounters one patch; 2. the parasitoid searches for hosts, lays its eggs and leaves this patch.

Therefore in each patch a specific time pattern Pi(t) of the number of searching parasitoids at time t arises. In order to determine this quantity we consider the parasitoids flying from one patch to another. In addition we introduce Po(t), the number of parasitoids searching for patches at time t. We use the probability todt that one parasitoid finds one patch in the time interval dt. Then we get

n

Po( t + d t ) = ( 1 - t o dt ) Po( t ) + E gito dtPo( t - Zi ), (17) i = 1

t

Pi(t) = • togiPo(t' ) d t ' , (18) I t = t - - Z i

n

e, = e 0 ( t ) + EPi(t). (19) i = 1

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After a short time an equilibrium Pi* is reached with

Pi* = Zigi°JP~

= n Pt 1 + w ~_ ,Z i (~ i )g i (a , )

i=1

Zi( oli) gi( a i ) n l° t .

T o + £ Z i ( o l i ) g i ( o l i ) i - I

/3i can be identified with

(20)

(21)

(22)

t,,* /~i-- p, (23)

T o is the mean time which a parasitoid spends searching for patches, and g,Z~ the mean time which a parasitoid spends searching for hosts in patch i. This equation connects the parasitoid response ~i to the host distribution as in a very simple way if the dependence of gi and Z, on a, is known. Together with this biological knowledge we are now able to calculate/~ for specific biological situations and to study its influence on the population dynamics. A schematic illustration of the parasitoid response Z(a~) to the patchily distributed host is shown in Fig. 2. The meaning of Fig. 2 is the following: small a values describe patches with a small part of the host population. In these patches the time Z(a) which a parasitoid stays is very short.

Because of the formal identity of Eqs. 12 and 16 we are able to use the full knowledge which arises from the stability conditions 13 and 14. By doing this we found two important results (Wiegand, 1992). We can confirm the known result (Hassell and May, 1974; Hassell, 1978) that stability increases with increasing aggregation of parasitoids and that stability requires the parasitoids not spending too much of their time in low host density regions. The second result is more surprising. We looked at the hidden asumptions of Eq. 16 and found that the

1.0

Z(~) 0.5

0.0 I

0.0 0.2 0.4

Fig. 2. A schematic illustration of the parasitoid response Z(a) to the patchily distributed host.

168 T. Wiegand, C. Wissel / Ecological Modelling 75 / 76 (1994) 161-170

concept of a i and /3 i presupposes the fractions ~i and J[~i to be the same in each generation. This premise is a good approximation if the host population is distributed in the same way each generation. This assumption is fulfilled if both hosts and parasitoids show a good migration ability. But this assumption is completely wrong if hosts are likely to stay in the patch where they hatched. In this case we get a very new form of dynamics.

3.2. A new form of dynamics

If hosts are likely to stay in the patch a new form of dynamic arises. Then we get an internal dynamics of hosts Ri, t within each patch i. These single dynamics are linked together by the parasitoids:

Ri, t+ 1 = LRi,t e-Ct3','P', (24) n

R t = E R i , t , (25) i=1

Ri , t+l = ( 2 6 ) with ai,t+l R t + l

and Pt+l=Rt(1- ~ e-Ct3i"P') (27)

By running this model (Wiegand, 1992) we find strong fluctuations in each patch and a total failure of the old stability concept. This deterministic equation leads to chaotic behaviour. Even the total population size R t and Pt are liable to fluctuations. We find coexistence if the system is able to smooth the strong fluctuations in the single patches. The main results are: 1. a minimum number of patches which actively participate in the dynamics is

required to reach coexistence; 2. parasitoids must avoid patches with very low host numbers (see Fig. 2); 3. the time T O which a parasitoid spends searching for patches acts as a stochastic

perturbation on the population dynamics. The longer To, the less stable is the host-parasitoid system.

4. C o n c l u s i o n

So far models describe host-parasitoid interaction in a phenomenological way. They consider the searching behaviour, which is considered in detail in this paper, only in a rough way. We introduce a model which describes the searching for hosts and laying of eggs of parasitoids in a spatial and temporal inhomogeneous environment more detailed and realistically. In a first step we concentrate on temporal aspects and investigate the influence of seasonal time patterns of available hosts and of parasitoids which are able to lay eggs on population

T. Wiegand, C. Wissel / Ecological Modelling 75/76 (1994) 161-170 169

dynamics. The model shows that a slight desynchronizat ion between host and parasi toid popula t ion leads to coexistence while perfect synchronizat ion causes instability. This gives rise to the conclusion that small environmental fluctuations p romote coexistence if they cause slight desynchronizat ion. We can identify the mechanism which supports stability. The system tends to be stable if a few hosts are likely to be unat tacked.

In a second step we concent ra te on spatial aspects, but we consider the searching behaviour as a dynamic process in space and time. F rom literature on empirical knowledge we extract decisive variables to describe the searching behaviour of parasitoids in a submodel. This submodel connects the parasitoid response /3 i to the host distribution ai in a very simple way. Toge ther with biological knowledge we are able to calculate /3i for specific biological situations and study its impact on populat ion dynamics. By doing this we could confirm the known result that stability increases with increasing aggregat ion of parasitoids. In addit ion we found that the existing hos t -paras i to id models are confined to very special cases. Hosts were assumed to have a good migrat ion ability. I f hosts are likely to stay in the patch where they hatch, a new form of dynamics arises. Then the model shows inherent fluctuations and strongly chaotic behaviour. We found coexistence if the system is able to smooth the strong host fluctuations in the single patches. This requires that parasitoids avoid patches with low host density and that the time which a parasitoids spends with searching for patches is relatively short.

Acknowledgement

This project was suppor ted by the Deutsche Forschungsgemeinschaf t (DFG).

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Host-parasitoid models in temporally and spatially varying environment