The effects of parasite-induced energetic costs on population dynamics in a consumer-resource model accounting for stage-structure.

The effects of parasite-induced energetic costs on

population dynamics in a consumer-resource model

accounting for stage-structure.

By: E. (Esmee) Bleker 11328886

Assessor: A. (André) de Roos Examinor: H. (Hal) Caswell Submission date: 03/07/2020

Abstract

Parasites play an important role in nature. They can affect their hosts in various ways. One of the most important ways in which parasites affect their hosts, is through energetic costs. Parasites use their hosts’ energy for their own growth and maintenance and hosts often also spend energy on immune responses and defence mechanisms. As a result, hosts have less energy to spend on growth or reproduction. In this study, the effects of energetic costs of parasitism on the population dynamics of the host, and also of other species which are in interaction with the host species, were examined. This study found that in the presence of a predator foraging on adult consumers and if juvenile consumers are stronger resource competitors, a parasite infecting juvenile competitors cannot invade the system if the energetic costs are low but can invade the system once the energetic costs cross a threshold value. In the absence of the predator, a parasite infecting adult consumers had a low optimum infection rate for which the density of infected individuals was highest. These counter-intuitive results show the importance of examining the effects of parasite-induced energetic costs in combination with the host stage structure. A deeper understanding of these effects might shed more light on the role of parasites in ecological communities.

Introduction

Parasites play an important role in nature, with parasitism being the most common consumer strategy (Lafferty et al, 2008). Occurring ubiquitously in nature, parasites increase species diversity (Lafferty et al, 2008), biomass (Kuris et al, 2008; Preston et al, 2013) and food-web complexity in ecological communities (Dunne et al, 2013; Cirtwill & Stouffer, 2015). Using data on the Carpinteria Salt Marsh in California, Lafferty et al. (2006) found that including parasites in the food web almost doubled its connectivity.

Parasites may also have a stabilising effect on ecological communities. Stable ecological communities possess three characteristics. The first of these is population dynamic stability, which means that the population rapidly reaches or returns to equilibrium, with possibly some mild fluctuations. The second characteristic is robustness to secondary extinction, which means that removing one species does not result in mass extinction of other species. The last characteristic is resilience, which means that the system quickly returns to its former state after perturbation (Hatcher, 2012). Properties expected to be associated with parasitism will generally increase stability (Poulin, 2010). Host-parasite interactions are often weak, as the effects of parasites on their hosts are often sublethal (Hatcher et al, 2006; Hatcher & Dunn, 2011). Strong interactions, like those between predator and prey, can have a destabilising effect on ecological communities. These effects may be mitigated by many weak interactions, like those between parasites and their hosts, enhancing the community’s dynamic stability. Parasites can therefore be expected to increase an ecosystem’s resilience and robustness to secondary extinction, by counteracting strong feedback relationships like those between predator and prey (Hatcher, 2012). Lafferty & Kuris (2009), however, found that parasites themselves are more vulnerable to secondary extinction and the addition of parasites could therefore lead to a decrease of a community’s robustness.

All life depends on energy, and trophic dynamics allow for energy transfer within an ecosystem (Lindeman, 1942). Energy enters the system through autotrophic organisms that, through photosynthesis, use solar energy to synthesize organic substances from inorganic substances. Heterotrophic organisms acquire energy by consuming these autotrophic organisms. Eventually, the tissue of dead organisms becomes the source of energy for saprophagous organisms (Thienemann, 1926). Some saprophagous bacteria transform the organic compounds in the tissue back into inorganic compounds, which can be used again by autotrophic organisms (Waksman, 1941).

Parasites also contribute to energy flows within ecosystems. They can be consumed by predators preying on their hosts, although the energetic benefits of the parasites to the predator are often small. Free-living stages can also be specifically preyed upon and could provide significant energetic resources for their predators (Johnson et al, 2010). Kuris et al (2008) examined three estuaries in California and found that parasite biomass exceeded that of top predators. But most importantly, parasites affect energy flows in ecosystems because they depend on their hosts for their metabolism (Booth, Clayton & Block, 1993), which hence incur energetic costs as a consequence of parasite infection. Furthermore, host individuals may invest energy into immune responses (Graham et al, 2011) or defence strategies (Rigby, Hechinger & Stevens, 2002), increasing the energetic costs induced by parasitism even further. These parasitism-induced energetic costs will reduce the amount of energy

host individuals can invest in growth and reproduction. Parasites can furthermore affect their hosts in various other ways, such as by castrating their hosts (Bonds, 2006) or changing their hosts’ behaviour to increase exposure to predation (Werner & Peacor, 2003).

Parasites do not only affect their host species but can also have an effect on populations of non-host species. They can, for example, cause apparent competition between host species that do not interact directly (Holt & Pickering, 1985). In this case, species exclusion might occur when one host species acts as disease reservoir for the other (Hatcher et al, 2012). Parasites can also be in competition with predators on their host species. The effects of parasites on non-host species are not always negative. If parasites regulate a dominant competitor more strongly than a non-dominant competitor, coexistence of both competitors is enhanced (Hatcher et al, 2006).

Anderson & May (1986) found that parasites that infect prey cannot invade a predator-prey system if the predators keep the predator-prey below a threshold density. The competition between parasite and predator can also result in extinction of the predator population, as the parasite can regulate the host population below the threshold density to sustain the predator population.

Packer et al (2003) examined a host-parasite model with predation. As predator population dynamics were not explicitly modelled, this model is more representative of predation by a generalist predator. They found that removal of the predator could lead to a decrease in host population size. This effect was strongest if predators selectively preyed on infected individuals, although it was also present if predators were non-selective.

Interactions between host and parasite are influenced by the developmental stage of both host and parasite, as demographic rates and infection dynamics differ. The influence of parasites is often felt more strongly in the older and larger individuals (Poulin, 2011), while smaller juvenile prey are more susceptible to predation (Hildrew et al, 2007). Host-parasite interactions can also lead to stage-specific biomass overcompensation. Stage-specific biomass overcompensation refers to the phenomenon that a change in demographic rates can relax density dependence in the life stage limiting biomass production most. For example, if mortality is low and juvenile consumers are weaker resource competitors than adult consumers, lack of sufficient resources for maturation will lead to a bottleneck in development. This population would consist mainly of juveniles. An increase in mortality would relax density dependence in the juvenile stage, allowing for a higher maturation rate, and thus a higher fraction of adults in the population (De Roos & Persson, 2013). Indeed Preston & Sauer (2020) found that juvenile biomass increased when adult mortality increased as a result of parasitism.

Extensive research has been done on the effects of parasitism, yet the energy drainage as a result of parasitism has not been accounted for in models of host-parasite interactions. This study is a step to remedying that omission by examining a simple model that accounts for maintenance costs, which will increase as a result of parasitism. The central research question to be addressed is how the energetic costs of parasitism affect the dynamics of the consumer and resource populations, as well as how parasitism can change the interactions between the consumer population and its predator.

Materials and methods

Four models were analysed in this study, all of which were consumer-resource models that explicitly account for stage structure in the consumer population and basic maintenance requirements of both juvenile and adult consumers. The first model incorporates a parasite that can infect only adult consumers. The second model incorporates a parasite than can only infect juvenile consumers. The last two models are extensions of the first two, where a predator on adult consumers is added to the model.

The stage-structured model incorporating adult parasitism is described by the following set of differential equations:

dR

dt

=

δ∗

(

R

max

−

R

)

−

a∗q∗R∗

(

C

a

+

I

a

)

−

a

q

∗

R∗C

j

d C

_a

dt

=

max

(

σ

_c

∗

a

q

∗

R−T

c

, 0

)

∗

C

j

−

μ

c

∗

C

a

−

i∗C

a

∗

I

a

d I

_a

dt

=

i∗C

a

∗

I

a

−

μ

c

∗

I

a

d C

j

dt

=

β

c

∗

max

(

σ

c

∗

a∗q∗R−T

c

, 0

)

∗

C

a

+

β

c

∗

max

(

(1−e)∗σ

c

∗

a∗q∗R−(1+d )∗T

c

, 0

)

∗

I

a

−

μ

c

∗

C

j

−

max

(

σ

c

∗

a

q

∗

R−T

c

, 0

)

∗

C

j

This model accounts for a resource population R that follows semi-chemostat dynamics,

δ (R

max

−

R)

in the absence of consumers with turn-over rate δ and

maximum resource density

R

max

.

Resource density decreases through consumption by

susceptible and infected adult consumers, as well as by juvenile consumers. Both juvenile and adult consumers forage following a linear functional response with attack rate

a /q

and aq , respectively. The parameter q therefore phenomenologically scales the ingestion rate of juvenile and adult consumer, which have equal foraging rates for

q=1

. For q>1 , adult consumers are better resource competitors than their juvenile counterparts, meaning the ingestion rate of juveniles is low. As maturation depends on the amount of energy individuals get from feeding, this causes a limitation in maturation. For

q<1

, juvenile consumers are better resource competitors than adults, meaning the ingestion rate of adults is low. Reproduction also depends on the amount of energy individuals get from feeding, so this causes a limitation in reproduction. The susceptible adult consumer population increases through maturation of juveniles (first term in the differential equation

dC

_a

/

dt

) and decreases through natural mortality (second term in

dC

_a

/

dt

) and infection by the parasite (last term in dCa/dt ). Maturation rate is represented by a

maximum function. The maturation rate depends on the available energy, which is given by the ingested food (functional response) multiplied with the efficiency with which consumers convert ingested food into energy (

σ

_c ), and the energy needed for essential, life-supporting processes ( Tc ). If individuals have less energy available than needed for these

essential processes, the maximum function sets the maturation rate to zero. The infected adult population increases through infection of susceptible adult consumers (first term in the differential equation

dI

a

/

dt

) and decreases through natural mortality (last term in

dIa/dt ). The juvenile consumer population increases through reproduction by susceptible

and infected adult consumers (first two terms in the differential equation

dC

j

/

dt

) and

decreases through natural mortality and maturation. The reproduction depends on the conversion efficiency

β

_c with which consumers can convert ingested food to net production. Reproduction rate, like maturation rate, is represented by a maximum function. The reproduction rate depends on the available energy, which is given by the ingested food (functional response) multiplied with the efficiency with which consumers convert ingested food into energy (

σ

c ) and for infected consumers also multiplied by the fraction of energy

not used by the parasite ( 1−e ). The reproduction rate furthermore depends on the energy needed for essential, life-supporting processes (

T

c for susceptible adults and

(1+d )∗T

c for infected adults due to energetic costs of immune system d). If individuals

have less energy available than needed for these essential processes, the maximum function sets the reproduction rate to zero. Hence, the parameters for parasite-induced energetic costs e and energetic costs of immune responses d, modify the energetics of infected adults by decreasing the amount of energy available for reproduction through decreasing energy

gained from foraging and increasing energy needed for other essential systems respectively. Natural mortality rate is assumed constant and equal to

μ

c for all host individuals,

irrespective of stage and infection status. The infection rate of susceptible adult consumers is proportional to the number of infected adults present with proportionality constant

i

.

The stage-structured model incorporating juvenile parasitism is described by the following set of differential equations:

C

(

¿¿

a+I

a

)−

a

q

∗

R∗(C

j

+

I

j

)

dR

dt

=

δ∗

(

R

max

−

R

)

−

a∗q∗R∗

¿

d C

_a

dt

=

max

(

σ

_c

∗

a

q

∗

R−T

c

, 0

)

∗

C

j

−

μ

c

∗

C

a

d I

_a

dt

=

max

(

(1−e)∗σ

_c

∗

a

q

∗

R−(1+d)∗T

c

, 0

)

∗

I

j

−

μ

c

∗

I

a

d C

_j

dt

=

β

c

∗

max

(

σ

c

∗

a∗q∗R−T

c

, 0

)

∗

C

a

+

β

c

∗

max

(

(1−e)∗σ

c

∗

a∗q∗R−(1+d )∗T

c

, 0

)

∗

I

a

−

max

(

σ

_c

∗

a

q

∗

R−T

c

,0

)

∗

C

j

−

μ

c

∗

C

j

−

i∗C

j

∗

I

j

−

i∗I

a

∗

C

j

d I

_j

dt

=

i∗C

j

∗

I

j

+

i∗C

j

∗

I

a

−

max

(

(1−e)∗σ

_c

∗

a

q

∗

R−(1+d )∗T

c

,0

)

∗

I

j

−

μ

c

∗

I

j

This model is a variation of the previous model. The equation describing the dynamics of the resource population R is mostly the same, except for an extra term incorporating the foraging of infected juveniles (last term in dR /dt ). Juveniles are now split into susceptible juveniles and infected juveniles. Dynamics of susceptible juveniles closely resembles the dynamics of juveniles in the previous model, increasing through reproduction of uninfected and infected adults and decreasing through natural mortality, however susceptible juveniles can now be infected by the parasite (last two terms in the differential equation dCj/dt ).

An equation for the dynamics of infected juveniles is added. This infected juvenile population increases through infection of susceptible juvenile consumers (first term in the differential equation

dI

_j

/

dt

) and decreases through maturation into infected adults (second term in dI_j/dt ) natural mortality (last term in dIj/dt ). Uninfected adults now come from the

maturation of susceptible juveniles only and can no longer be infected. Infected adults now increase through maturation of infected juveniles instead of by infection of susceptible adults (first term in

dI

_a

/

dt

).

The stage-structured model incorporating adult parasitism and a predator population foraging on adult consumers is described by the following set of differential equations:

dR

dt

=

δ∗

(

R

max

−

R

)

−

a∗q∗R∗

(

C

a

+

I

a

)

−

a

q

∗

R∗C

j

d C

_a

dt

=

max

(

σ

_c

∗

a

q

∗

R−T

c

, 0

)

∗

C

j

−

μ

c

∗

C

a

−

i∗C

a

∗

I

a

−

b∗C

a

∗

P

d I

a

dt

=

i∗C

a

∗

I

a

−

μ

c

∗

I

a

−

b∗I

a

∗

P

d C

_j

dt

=

β

c

∗

max

(

σ

c

∗

a∗q∗R−T

c

, 0

)

∗

C

a

+

β

c

∗

max

(

(1−e)∗σ

c

∗

a∗q∗R−(1+d )∗T

c

, 0

)

∗

I

a

−

μ

c

∗

C

j

−

max

(

σ

_c

∗

a

dP

dt

=

β

p

∗

max

(

σ

p

∗

b∗(C

a

+

I

a

)−

T

p

, 0

)

∗

P−μ

p

∗

P

This model is an extension of the first model. The equations for resource density and juvenile consumer density remain unchanged. For susceptible and infected adults there is an additional mortality term (the last term in

dC

a

/

dt

and

dI

a

/

dt

respectively). Lastly, there

is an additional equation for the dynamics of the predator that forages on adult consumers. This predator population increases through foraging on susceptible and infected adult consumers (first term in the differential equation dP/dt ) and decreases through natural mortality (last term in

dP

_j

/

dt

). The reproduction depends on the conversion efficiency

β_p with which predators can convert ingested food to net production.

The stage-structured model incorporating juvenile parasitism and a predator population foraging on adult consumers is described by the following set of differential equations:

C

(

¿¿

a+I

a

)−

a

q

∗

R∗(C

j

+

I

j

)

dR

dt

=

δ∗

(

R

max

−

R

)

−

a∗q∗R∗

¿

d C

_a

dt

=

max

(

σ

_c

∗

a

q

∗

R−T

c

, 0

)

∗

C

j

−

μ

c

∗

C

a

−

b∗C

a

∗

P

d I

_a

dt

=

max

(

(1−e)∗σ

_c

∗

a

q

∗

R−(1+d)∗T

c

, 0

)

∗

I

j

−

μ

c

∗

I

a

−

b∗I

a

∗

P

d C

_j

dt

=

β

c

∗

max

(

σ

c

∗

a∗q∗R−T

c

, 0

)

∗

C

a

+

β

c

∗

max

(

(1−e)∗σ

c

∗

a∗q∗R−(1+d )∗T

c

, 0

)

∗

I

a

−

max

(

σ

_c

∗

a

q

∗

R−T

c

,0

)

∗

C

j

−

μ

c

∗

C

j

−

i∗C

j

∗

I

j

−

i∗I

a

∗

C

j

d I

_j

dt

=

i∗C

j

∗

I

j

+

i∗C

j

∗

I

a

−

max

(

(1−e)∗σ

c

∗

a

q

∗

R−(1+d )∗T

c

,0

)

∗

I

j

−

μ

c

∗

I

j

dP

dt

=

β

p

∗

max

(

σ

p

∗

b∗(C

a

+

I

a

)−

T

p

, 0

)

∗

P−μ

p

∗

P

This model is an extension of the second model presented above. The equations for resource density, susceptible juvenile consumer density and infected juvenile consumer density remain unchanged. For uninfected and infected adults there is an additional mortality term (the last term in dCa/dt and dIa/dt respectively). Lastly, there is an additional

equation for the predator foraging on adult consumers. Predator population dynamics are the same as for the previous model.

Model analysis

The possible equilibrium states of the models without predation were investigated as a function of the infection rate parameter i, whereas the possible equilibrium states of the models with predation were investigated as a function of the infection rate i and the parasite-induced energetic costs e. Lastly, a two-parameter bifurcation analysis as a function of infection rate and energetic costs was carried out for the model incorporating a parasite infecting juvenile consumers and a predator foraging on adult consumers. All these analyses were carried out for the situation where juvenile consumers are stronger resource competitors than adult consumers. All bifurcation analyses were carried out in R, using the deBif package.

Graph 1 shows the default parameter values.

Results

Model with parasites infecting adult consumers

Figure 1 shows the possible equilibrium states of the model without predators and with parasites that infect adult consumers for different values of the infection rate. A threshold value of the infection rate is found at i = 0.00221, which is referred to as a branching point. To the left of the branching point the parasite cannot invade the system, and to the right of the branching point the parasite can invade the system. With increasing infection rate the resource density increases before levelling off, after which it increases very slightly. This kink in the curve results from the resource density having increased enough for infected adult consumers to reproduce. With infected adult consumers unable to reproduce, the consumer population declines allowing the resource density to increase. Once infected adult consumers can reproduce, the consumer population declines only slightly with further increases in infection rate so the resource does not increase much further either. Juvenile consumer density drops with increasing infection rate, before levelling off after which it decreases very slightly. The drop in juvenile density is due to the decline in susceptible adults leading to a decline in reproduction. Reproduction rate of susceptible adults increases due to increased resources, but their density strongly decreases, negatively affecting juvenile density. Once infected adults start reproducing, this decline in juvenile density is halted. Susceptible adult consumer density decreases rapidly with increasing infection rate, before slowly levelling off after which it decreases very slightly. This is due to susceptible adults becoming infected by the parasite. Initially infected adult consumer density rapidly increases, reaching a peak density. As infection rate continues to increase the infected adult consumer density decreases before levelling off. This optimum of infected adult density is at a very low infection rate. As a result of this low infection rate newly matured adults still have time to reproduce before becoming infected, as reproduction and maturation rate have increased due to an increase in resource density. As infection rate increases further, however, fewer adults will be able to reproduce before becoming infected, resulting in a decline in consumer population density which is most starkly seen in the infected adult consumer density. Eventually, the consumer population drops enough for the resource to reach a density at which infected adult consumers can also reproduce.

Figure 1: Parameter bifurcation graph as a function of infection rate i for the consumer-resource model with adult parasitism. This figure shows the stable equilibrium of the resource density, juvenile consumer density, susceptible adult

consumer density and infected adult consumer density with increasing infection rate. A branching point is found at an infection rate of 0.00221, above which value the infection can persist in the host population.

Model with parasites infecting juvenile consumers

Figure 2 shows a bifurcation over infection rate i for the model, in which parasites infect juvenile consumers. A branching point is found at i = 0.12509. To the left of the branching point the parasite cannot invade the system, and to the right of the branching point the parasite can invade the system. This infection rate threshold for which the parasite can invade the system is much higher than for a parasite infecting adult consumers. This is most likely due to the low density of juvenile consumers in the system, as their maturation rate is high due to them being better resource competitors.

With increasing infection rate the resource density increases as a result of the decline in consumer population density due to infected adult consumers not being able to reproduce.

Susceptible juvenile consumer density drops with increasing infection rate, as they become infected by the parasite. Infected juvenile consumer density increases with increasing infection rate before starting to slowly decrease. This optimum curve in the infected juvenile density most likely results from the combination of an increasing infection rate and the maturation rate being positively affected by the increase in resource density. Initially, infected juvenile consumer density increases because of the increase in the infectivity parameter. Maturation, however, increases as a result of increasing resources, which leads to a peak in infected juvenile consumer density followed by a decrease in infected juvenile consumer density. Uninfected adult consumer density decreases with increasing infection rate, as susceptible juvenile density decreases. Infected adult consumer density increases with increasing infection rate because the infected juvenile density increases. While there is an optimum curve in the infected juvenile density, this system does not have an optimum in the total number of infected individuals as seen in the previous system as infected adult density continues to increase.

Figure 2: Parameter bifurcation graph as a function of infection rate i for the consumer-resource model with juvenile parasitism. This figure shows the stable equilibrium of the resource density, susceptible juvenile consumer density,

infection rate. A branching point is found at an infection rate of 0.12509, above which value the infection can persist in the host population.

Model with parasites infecting adult consumers and predators foraging on adult consumers

Figure 3 shows the possible equilibrium states of the model with predators and parasites that infect adult consumers for different values of the infection rate. A branching point is found at i = 0.05725. To the left of the branching point the parasite cannot invade the system, and to the right of the branching point the parasite can invade the system. The infection rate threshold for which the parasite can establish itself is higher than in a system without the predator. This increase in the invasion threshold is due to the increased mortality as a result of predation and a decreased density of susceptible adults. As such, an infected individual has a shorter lifespan which means that it can infect fewer other individuals, and there are fewer susceptible individuals to get infected. The combination of these factors leads to the parasite needing a higher infection rate in order to invade.

With increasing infection rate the resource density increases before levelling off and increasing only very slightly afterward. This is due to the consumer population decreasing with increasing infection rate as infected adults cannot reproduce until the resource density has sufficiently increased. The kink is due to the resource reaching a high enough density for infected adults to reproduce. Juvenile consumer density decreases with increasing infection rate before levelling off and decreasing only very slowly. Like in the system without the predator, this is due to the decrease in reproduction as a result of the decreasing density of susceptible adults and the inability of infected adults to reproduce. The kink is again caused by the infected adults having enough resources to start reproducing. Susceptible adult consumer density decreases with increasing infection rate before levelling off and decreasing only very slowly. This is again due to susceptible adults getting infected by the parasite. Infected adult consumer density increases with increasing infection rate before levelling off and increasing only very slightly. The increase is a result of susceptible adults becoming infected. The kink is again due to infected adults being able to reproduce. Predator density decreases before levelling off and decreasing only very slightly. This decrease is due to the decrease in juvenile density. As a result, there is less maturation of consumers which leads to less food being available for the predator.

The optimum of infected individuals seen in absence of the predator, is not present in this system, as the total adult consumer density is top-down controlled by the predator and hence does not change with infection rate. Instead, the infected adult consumer density increases with increasing infection rate until eventually all individuals become infected upon maturation and the infected adults make up the entire adult population. The predator, however, does have fewer prey available which results in a decline of the population.

Figure 3: Parameter bifurcation graph as a function of infection rate i for the consumer-resource model with adult parasitism and a predator population foraging on adult consumers. This figure shows the stable equilibrium of the

resource density, juvenile consumer density, susceptible adult consumer density, infected adult consumer density and predator density with increasing infection rate. A branching point is found at an infection rate of 0.05725, above which value the infection can persist in the host population.

Figure 4 shows the possible equilibrium states of the model with predators and parasites that infect adult consumers for different values of parasite-induced energetic costs. With increasing costs the resource density increases before levelling off. The increase in resource density is due to the decline of the consumer population. Once the energetic costs get too high for infected adults to reproduce, the consumer population no longer declines so the resource density stops increasing. Juvenile consumer density decreases with increasing energetic costs before levelling off. This decrease in juvenile density is a result of the decrease in reproduction of infected adults due to energetic costs. The juvenile density stops decreasing when infected adults can no longer reproduce. Susceptible adult consumer density decreases with increasing energetic costs. The decrease in juvenile density leads to less maturation. This results in a drop in predator density as there is less food available. The lower predation increases the lifespan of infected adults, allowing them to infect more susceptible adults, thereby decreasing the susceptible adult density. Infected adult consumer density increases with costs as more susceptible adults become infected. Predator density decreases before levelling off. Food availability drops with decreasing juvenile density, leading to a decline of the predator population. Once infected adults stop reproducing, juvenile density stops decreasing, allowing the predator population density to level off as well.

Figure 4: Parameter bifurcation graph as a function of parasite-induced energetic costs e for the consumer-resource model with adult parasitism and a predator population foraging on adult consumers. This figure shows the

stable equilibrium of the resource density, juvenile consumer density, susceptible adult consumer density, infected adult consumer density and predator density with increasing infection rate.

Model with parasites infecting juvenile consumers and predators foraging on adult consumers

Figure 5 shows the possible equilibrium states of the model with predators and parasites that infect juvenile consumers for different values of the infection rate. A branching point is found at i = 0.34276. To the left of the branching point the parasite cannot invade the system, while to the right of the branching point the parasite can invade the system. As in the previous system, the infection rate threshold for parasite invasion is higher in presence of the predator than in its absence. The density of susceptible juveniles is higher when the predator is present, but the resource density is also much higher. This means that the maturation rate is much higher than in absence of the predator so there is less time in which a juvenile can be infected. This is the most likely explanation of the higher threshold value for invasion.

With increasing infection rate the resource density increases as the consumer population decreases. Susceptible juvenile consumer density decreases with increasing infection rate as they become infected by the parasite and as a result of infected adults not having sufficient resources available for reproduction. Infected juvenile consumer density increases with increasing infection rate as more juveniles become infected. Uninfected adult consumer

density decreases due to the decrease in susceptible juveniles. Infected adult consumer density increases with increasing infection rate as there are more infected juveniles. Predator density decreases with increasing infection rate due to the decrease in juvenile consumer density which leads to a decrease in maturation and thus the predator has less food available.

Figure 5: Parameter bifurcation graph as a function of infection rate i for the consumer-resource model with juvenile parasitism and a predator population foraging on adult consumers. This figure shows the stable equilibrium of the

resource density, susceptible juvenile consumer density, infected juvenile consumer density, uninfected adult consumer density, infected adult consumer density and predator density with increasing infection rate. A branching point is found at an infection rate of 0.34276, above which value the infection can persist in the host population.

Figure 6 shows the possible equilibrium states of the model with predators and parasites that infect juvenile consumers for different values of the parasite-induced energetic costs. An equilibrium without parasites present occurs for low parasite-induced energetic costs up to a branching point found at e = 0.66549. Another equilibrium with parasites present occurs at high parasite-induced energetic costs above e = 0.71934, where a second branching point is found. To the left of the first branching point at e = 0.66549, the parasite cannot invade the system. In between both branching points, there is bistability, with the parasite either being absent from or present in the system. To the right of the second branching point, the parasite is always present.

In the two different types of stable equilibrium states occurring at low and high parasite-induced energetic costs, respectively, the density of infected adult consumers is always 0 and the susceptible adult consumer density is top-down controlled by predators at the same

level. Only in the bistability region between the two bifurcation points at e = 0.66549 and e = 0.71934 the two stable equilibrium states are separated from each other by an unstable equilibrium state (saddle point), in which the infected adult consumer density is positive. Given that it is unstable this saddle point will not be discussed further here. When parasites are present they only occur in infected juvenile consumers and they lead to lower densities of resources, susceptible juvenile consumers and predators. Once the parasites are present, increases in the parasite-induced energetic costs do not lead to further changes in the equilibrium state. This results from the maximum curve setting the maturation rate of infected juveniles to zero, as the maturation rate under these conditions would be negative. As parasite-induced energetic costs increase, the maturation rate also decreases but as maturation cannot be negative, the maximum function keeps the maturation rate set zero. Hence, increasing the parasite-induced energetic costs has no effect.

The area of bistability over infection rate and energetic costs is indicated in area 4 of figure 7. Area 1 and 3 of this graph indicate for which values of infection rate and energetic costs the parasite can persist. The values of infection rate and energetic costs for which the parasite cannot persist are indicated by area 2 and area 5.

Figure 8 shows the R0 of the equilibrium without the parasite present and with the

parasite present as a function of energetic costs. The rate with which infected juveniles and adults infect susceptible juveniles is

i∗C

j . Following infection a juvenile can either die or

mature. Hence, the amount of susceptible juveniles an infected juvenile infects is given by

i∗C

_j

μ

c

+

maturation

. The probability that an infected juvenile matures into an infected adult is

maturation

μ

_c

+

maturation

. Following maturation, the infected adult can either die naturally or die

due to predation. Hence, the amount of susceptible juveniles an infected adult infects is given by

_b∗P+μ

i∗Cj

c

. R0 is then given by the sum of the amount of susceptible juveniles

infected by an infected adult multiplied by the probability of maturation of an infected juvenile and the amount of susceptible juveniles infected by an infected juvenile:

i∗C

_j

∗(

1

μ

c

+

maturation

+

maturation

(

μ

c

+

maturation

)

∗(

b∗P+μ

c

)

)

_{, which can also be written as}

i∗C

j

∗(

b∗P+μ

c

+

maturation)

(

μ

c

+

maturation

)

∗(

b∗P+μ

c

)

.

This graph shows that the area of bistability can be explained with the R0 value. As

stated, a parasite can persist in the system if the R0 value is 1 or higher. In this model the R0

value increases as maturation decreases. This is a result of the higher mortality rate in the adult stage due to predation. When the maturation rate decreases as a result of infection, juveniles live longer in the juvenile stage where they are safe from predation. As such, infected individuals have more time to infect other individuals when they spend more time as juveniles. The energetic costs negatively affect maturation of infected juvenile consumers, meaning that the R0 value increases with increasing costs. Figure 9 illustrates that the R0 of

the parasites in the equilibrium in which they are present drops below 1 for values of the energetic costs to the left of the left-most branching point from Figure 7 – the minimum value of parasite-induced energetic costs at which the parasite can persist in the system. The R0

value of the parasites in the equilibrium from which they are absent exceeds 1 for values of the energetic costs to the right of the right-most branching point from Figure 7 – the minimum value of parasite-induced energetic costs above which there is always persistence of the parasite. Between both branching points the R0 of the equilibrium with the parasite is above 1

while the R0of the equilibrium without the parasite is below 1, thus creating a region of

Figure 6: Parameter bifurcation graph as a function of parasite-induced energetic costs e for the consumer-resource model with juvenile parasitism and a predator population foraging on adult consumers. This figure shows two

stable equilibria of the resource density, juvenile consumer density, susceptible adult consumer density, infected adult consumer density and predator density with increasing infection rate, as well as an unstable equilibrium between two branching points. The first branching point in this graph is found at a value for parasite-induced energetic costs of 0.66549. The second branching point in this graph is found at a value for parasite-induced energetic costs of 0.71934. Between these branching points is a region of bistability.

Figure 7: Two-parameter bifurcation plot varying both parasite-induced energetic costs and infection rate.

This figure shows the location of the left-most branching point found in figure 6 (red line) and the location of the right-most branching point found in figure 6 (blue line) as a function of infection rate and parasite-induced energetic costs. In area 1 and 3 there is persistence of the parasite. In area 2 and 5 there is extinction of the parasite. In area four there is bistability, where the parasite can either persist or go extinct.

2

1

3

5

4

Figure 8: R0 of the parasites in the two stable equilibrium states found in the consumer-resource model with

juvenile parasitism and a predator population foraging on adult consumers as a function of energetic costs.

The red line in this figure shows the R0 value of the parasites in the equilibrium state in which they are present. This line drops

below 1 at the left-most branching point in Figure 6, which was found at a value for parasite-induced energetic costs of 0.66549. The blue line indicates the R0 value of the parasites in the equilibrium from which they are absent. This line crosses 1 at the

right-most branching point in Figure 7, which was found at a value for parasite-induced energetic costs of 0.71934. The vertical grey lines indicate the e-values at which the branching points were found. The horizontal grey line indicates an R0 value of 1.

Discussion

The central research question of this study was how the energetic costs of parasitism affect the dynamics of the consumer and resource populations, and the interactions between the consumer population and its predator. The most notable results were the different effects of a parasite infecting adult consumers versus a parasite infecting juvenile consumers, in a consumer-resource model where juvenile consumers are stronger resource competitors. Two results specifically were very interesting. The first of these was the optimum infection rate found for a parasite infecting adult consumers in a system without predation, which was not found for a parasite infecting juveniles. This optimum was the result of a very low infection rate, which allowed newly matured adults to have time for reproduction before getting infected with the parasite. Such an optimum of infected individuals was not present for a parasite which only infects juveniles. There was an optimum in the infected juvenile density, caused by the interplay between the increasing infectivity and the increase in maturation rate

as a result of the increase in resource density. Initially, infected juvenile consumer density increases because of the increase in the infectivity parameter. Maturation, however, increases as a result of increasing resources, which leads to a peak in infected juvenile consumer density followed by a decrease in infected juvenile consumer density.

However, as the infected adult density continued to increase, a parasite infecting juveniles does not have an optimum infection rate like a parasite infecting adults does. The second interesting result was that a parasite infecting juveniles could only invade a system with predation if its energetic costs were high, which was not found for a parasite infecting adult consumers. This resulted from the R0 value being positively affected by a decrease in

maturation, as juveniles are safe from predation and thus have a lower mortality than adults. A parasite infecting adults cannot use this strategy to increase its R0 value, which instead

does not change with increasing energetic costs.

This study also found that the predator and the parasite had negative indirect effects on each other. Previous research found that the competition between parasite and predator could lead to predator extinction if the parasite regulates the host below the threshold density to sustain the predator, but also that the predator may regulate the prey population below the threshold value for parasite invasion (Anderson & May, 1986). Indeed, this study found that in a system where the consumer population is regulated by a predator, a parasite needs to be more infective to be able to invade than it would in a system without the predator. For the parasite infecting adult consumer this is most likely due to the reduced amount of susceptible individuals. For the parasite infecting juveniles, the amount of susceptible individuals is actually higher in a system with predation. However, the total consumer population density is lower, allowing for a higher resource density. This results in an increased maturation speed. This rapid maturation in combination with a higher mortality in the adult stage, reduces the time in which an infected individual can infect others.

The parasite also negatively affected the predator. Both a parasite infecting adults and a parasite infecting juveniles negatively affected the predator population density when their infectivity increased. This is the result of a decline in the juvenile population due to infected adults not having sufficient energy for reproduction. As a result, there is less food for the predator population, leading to its decline. Both parasites also negatively affected the predator population when their energetic costs increased, as this also led to a decline in the juvenile population. For a parasite infecting adults, this juvenile decline is due to the energetic costs negatively affecting the reproduction of infected adults. For a parasite infecting juveniles, however, this is due to the energetic costs preventing infected juveniles from maturing into adults. In this case, the parasite cannot persist at low energetic costs, but at high energetic costs the susceptible juvenile density decreases as they get infected by the parasite. The infected juveniles do not mature, so this also leads to a reduction in the amount of food available to the predator population.

Previous research into the effects of parasites on population dynamics have focussed on increased mortality as a result of infection, as parasites may change their hosts’ behaviour to increase exposure to predation (Werner & Peacor, 2003). The effects of the energetic costs of parasitism on the dynamics in a consumer-resource system have not often been taken into account. Increased mortality resulting from parasitism has a decidedly different effect on R0,

which depends on the number of susceptible individuals in the surroundings and the time an infected individual has to infect new individuals. A parasite which increases its host’s mortality, negatively affects the time an infected individual has to infect new individuals which means that it decreases its R0 value. The effect of parasite-induced energetic costs on the R0

value is quite different. The R0 value of a parasite infecting adult consumers is not affected

by the energetic costs the parasite imposes on its host, in a system where juveniles are stronger resource competitors and there is a predator population which forages on adult consumers. As the energetic costs increase, there is an increase in the density of infected individuals, but this is not the result of an increase in R0. Instead, this is caused by the

decline in the predator population as a result of the reduction of reproduction by infected adult consumers. This predator decline leads to a lower mortality due to predation, allowing adult consumers to live longer thereby increasing the time an infected individual has to infect

new individuals. However, the predator population does keep the total adult density constant, so the amount of susceptible individuals decreases as they become infected, cancelling out the extra time individuals have to infect new individuals. More interesting is the effect of parasite-induced energetic costs of a parasite infecting juveniles on the R0 value. In this

system, where adult consumers have a higher mortality due to predation, infected individuals can increase the time they have to infect new individuals by spending more time in the juvenile stage where they are safe from predation. In this system, the number of susceptible individuals also decreases, but not as severely as in a system with a parasite infecting adults as juveniles are not being regulated by a predator. Therefore, by decreasing the maturation rate, a parasite infecting juvenile consumers can increase its own R0 value.

Energetic costs are a key characteristic of parasitism, as parasites are dependent on their hosts for their metabolism (Booth, Clayton & Block, 1993). Often hosts will also invest energy into immune responses (Graham et al, 2011) and defence strategies (Rigby, Hechinger & Stevens, 2002). This study has shown that energetic costs incurred by parasitism can affect population dynamics in counterintuitive ways. Therefore it is important to take these costs into account when examining the effects of parasitism on population dynamics. With parasitism being the most common consumer strategy (Lafferty et al, 2008), a deeper understanding of its effects on the host and its indirect effects on non-host species is needed. As of yet, the effect of parasitism on ecological communities is unclear, with speculation that parasitism may increase a community’s stability (Poulin, 2010) as well as speculation that it may decrease stability (Lafferty & Kuris, 2009). A deeper look into how energetic costs of parasitism may affect the host and its interactions with other species, might shed more light on the role of parasitism in ecological communities.

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