Food web stability: the interplay among switching predation behavior,

interaction strength and environmental heterogeneity

Lucas Del Bianco Faria§

Michel Iskin da Silveira Costa§

§Laboratório Nacional de Computação Científica, Av. Getúlio Vargas, 333 –

Quitandinha, Petrópolis (RJ), 25651-070 – Brazil. Phone: +55 (24) 2233 6008, Fax: +55

(24) 2233 6141.

E-mail: lucasbf@lncc.br and michel@lncc.br

______________________________________________________________________

Abstract

Food webs are a central idea in ecology and an useful departure point for the

development of a predictive theory of community ecology. Empirically ecological

research has demonstrated that food webs may contain a vast array of species connected

via multiple links of variable interaction strength. Within this context, this work

investigates the role that interaction strength, environmental heterogeneity and structure

of prey consumption play in the determination of some food web dynamics.

Key words: food web theory, interaction strength, nutrient enrichment, apparent

competition

1 Introduction

Food webs represent the trophic connections among taxa in ecological

communities, indicating energy path and nutrient flow through living organisms.

Empirically ecological research has amply demonstrated that food webs may contain a

vast array of species connected via multiple links named interactions. As a relevant

character of these interactions, it is commonly found in nature that predators have a

wide diet range, consuming more than one prey item (e.g., generalist predator,

intraguild predation and omnivory). In this light, studies have been performed to assess

the effect of prey selection (preference) on consumer-resource dynamics, spurring a

lively theoretical debate about the relation among species interaction strength, diversity

and stability in food webs (McCann and Hastings, 1997; Huxel and McCann, 1998;

Huxel et al., 2002). A common feature of these works is prey preference structure

which, according to the cited references, can be embedded in two settings (sensu Holt,

1983) from the predation point of view: (1) non-switching predators; (2) switching

predators. The first setting portrays a fixed predator behavior with respect to the density

of its alternative prey. The second is based upon the relative prey densities, within the

frame of predator switching sensu Murdoch and Oaten (1975). Associated to it is also

the concept of risk index, proposed by some authors (Matsuda, 1986) to refer to the

switching behavior of the predator. This index states that a prey is more prone to

predation as its density increases with respect to other alternative prey. These trophic

interactions together with their strength may be of great importance on account of the

diffusion of consumption and productivity effects across trophic levels, consequently

increasing food web connectance (Polis, 1991; Strong, 1992). Several studies have

suggested that weak to intermediate strength links are important in promoting

community persistence, whereas weak links act to dampen oscillations between

consumer and resource (McCann and Hastings, 1997; McCann et al., 1998; Huxel and

McCann, 1998; Post et al., 2000; Teng and McCann, 2004).

In this work, for two specific consumer-resource models, namely, apparent

competition, apparent competition/shared predator, it is shown that the presented

preference structures can have a marked effect on the ultimate consumer-resource

dynamics. This may point to the importance of the choice of the prey selection structure

in the analysis of diversity-stability issues in food web theory.

3.2 Consumer-resource dynamics models and prey preference terms

The structures of the models to be analyzed consist of two non-interacting prey,

N1, N2, which share a predator, G, as shown in figure 1A, and the same two non-

interacting prey, sharing a predator, G and a specialist predator, S, which preys on N1, as

shown in figure 1B. Case (B) encompasses two types of food webs, namely, shared

predator and shared prey.

Figure 1. System A - two non-interacting prey sharing a predator; system B - two non-interacting prey sharing a predator and a specialist predator on N1 prey. The predator consumption rate may either obey a switching or a non-switching behavior.

G

N2 N1

(1-p) p

N2 N1

S

G

(A) (B)

p (1-p)

Within the non-switching framework, the models for the setting in figure 1A and

1B with prey selection structure presented in (Faria et al., 2007) (hereinafter named

non-switching: NSW – A and NSW – B) have the following form, respectively:

1 1 11 1

1 1 2

(1 )1

1 (1 ) h h

dN N p aGN= r N

dt K + p aT N + paT N

−− − −

2 2 22 2

2 1 2

11 (1 ) h h

dN N paGN= r N

dt K + p aT N + paT N

− − −

(NSW - A)

1 2

1 2

(1 )

1 (1 )gh h

p aGN + paGNdG= c

dt + p aT N + paT N

− −

111 11 1

1 1 1 2

(1 )1

1 1 (1 )gs

s h g h g hs g g

p a GNa SNdN N= r N

dt K +a T N + p a T N + pa T N

− − − − −

22 22 2

2 1 2

11 (1 )

g

g h g hg g

pa GNdN N= r N

dt K + p a T N + pa T N

− − −

1 2

1 2

(1 )

1 (1 )g g

g gg h g hg g

p a GN + pa GNdG= c d G

dt + p a T N + pa T N

− −

−

(NSW - B)

1

11s

s ss hs

a SNdS= c d S

dt +a T N

−

while for the switching framework, with the selection structure presented in Post et al.

(2000) (hereafter named switching: SW – A and SW – B), the models take on the form:

11 1 11 1

1 1 2 1

(1 )1

(1 ) 1g

g

g h

a GNdN N p N= r N

dt K p N pN +a T N

−− − − +

22 2 22 2

2 1 2 2

1(1 ) 1

g

g

g h

a GNdN N pN= r N

dt K p N pN +a T N

− − − +

(SW - A)

1 21 2

1 2 1 1 2 2

(1 )

(1 ) 1 (1 ) 1g g

g gg g

g h g h

a GN a GNp N pNdG= c d G

dt p N pN +a T N p N pN +a T N

− + − − + − +

111 1 11 1

1 1 1 2 1

(1 )1

1 (1 ) 1s g

gs

s h g h

a GNa SNdN N p N= r N

dt K a T N p N pN +a T N

−− − − + − +

22 2 22 2

2 1 2 2

1(1 ) 1

g

g

g h

a GNdN N pN= r N

dt K p N pN +a T N

− − − +

(SW - B)

1 21 2

1 2 1 1 2 2

(1 )

(1 ) 1 (1 ) 1g g

g gg g

g h g h

a GN a GNp N pNdG= c d G

dt p N pN +a T N p N pN +a T N

− + − − + − +

1

11s

s ss hs

a SNdS= c d S

dt +a T N

−

G and S represent the generalist and the specialist predator, respectively, while N1

denotes the pest prey and N2 the endemic prey; ri and Ki (i = 1, 2) are the prey growth

rates and carrying capacities, respectively; ci (i = g and s) are the efficiency conversion

of prey biomass into predators growth; ai (i = g and s) are the predators attack rate and

Thi (i = g and s) are the predators handling time; di (i = g and s) is the predator mortality

rate. In all models the parameter p (0 ≤ p ≤ 1) is the preference term for prey

consumption. For instance, if p = 1 preference is directed entirely to prey N2 (p is also

called bias in switching response (Matsuda et al., 1986)).

The objective of the analysis is twofold: it deals with the assessment of the

dynamics of the above models under variation of the selection term p so as to highlight

the possible influence of interaction strength; it provides an assessment of the dynamics

under variation of the environmental productivity conveyed here by the ratio K1/K2 (this

specific analysis is restricted to system A in Fig 1). The reason for the second item

draws on the fact that prey species frequently experience different productivities in the

environment, which may alter significantly the food web dynamics.

3.3 Results

For the preferences structures of models (NSW - A) and (SW - A) a bifurcation

diagram with local maxima and minima for each species was built as a function of the

selection term p (0 ≤ p ≤ 1). The procedure of the investigation consisted of running

both systems (NSW and SW) for 10000 iterations, and examining only the last 5000

time steps (to eliminate transient phase), where local long-term maxima and minima

were plotted.

Figure 2A-C (NSW - A) shows that for the non-switching selection term, a

stable system (i.e., pest extinction) can be attained for low values of p (approximately, 0

< p ≤ 0.4), while destabilization (i.e., endemic extinction) occurs for high values of p

and around 0.5 (i.e. maximum degree of predator generalism).

As for the switching selection structure in model (SW - A), figure 3 A-C

suggests that limit cycles occur for the whole set of values of p. Besides, trajectories

seem to pass near zero, increasing thus the system instability.

Figure 2. Bifurcation diagrams as a function of the choice parameter p for NSW – A: models: (A) prey 1 for model; (B) prey 2 for model; (C) generalist predator for model. Parameter values: r1 = 1.0; r2 = 1.0; K1 = 4.0; K2 = 4.0; ci = 0.3; Thi = 0.5; di = 0.1; ai = 1.0.

Figure 3. Bifurcation diagrams as a function of the choice parameter p for SW – A: models: (A) prey 1 for model; (B) prey 2 for model; (C) generalist predator for model. Parameter values: r1 = 1.0; r2 = 1.0; K1 = 4.0; K2 = 4.0; ci = 0.3; Thi = 0.5; di = 0.1; ai = 1.0.

Likewise, for the non-switching and switching preference structures of model B

(NSW – B and SW – B) a bifurcation diagram with local maxima and minima for each

species was built as a function of the selection term p (0 ≤ p ≤ 1).

At first glance, it is evidenced that for both structures, specialist predator can

invade and persist only when the generalist predator begins to turn its preference to the

endemic pest species (i.e., 0.5 ≤ p).

On the other hand, the dynamics of both models differ significantly: the non-

switching structure oscillates between two points, experience complex behavior and

return to two-point limit cycle; in turn, the switching structure displays a two-point limit

cycle for all values of p with all populations attaining very low levels, which points to a

likely all-species extinction case (i.e., general instability).

A cross-model comparison shows that the addition of a specialist does not

influence the dynamical results pertaining to a generalist only as long as p < 0.5 (cf.

Figs.2 and 4). However, beyond that limit value of p (0.5 < p < 0.8) complex behavior

ensues for the specialist-generalist setup, as opposed to the stable and two-point limit

cycle behavior of the generalist setup.

Figure 4. Bifurcation diagrams as a function of the choice parameter p for NSW – B: models: (A) prey 1; (B) prey 2; (C) generalist predator; (D) specialist predator. Parameter values: r1 = 1.0; r2 = 1.0; K1 = 4.0; K2 = 4.0; ci = 0.3; Thi = 0.5; di = 0.1; ai = 1.0.

Figure 5. Bifurcation diagrams as a function of the choice parameter p for SW – B: models: (A) prey 1; (B) prey 2; (C) generalist predator; (D) specialist predator. Parameter values: r1 = 1.0; r2 = 1.0; K1 = 4.0; K2 = 4.0; ci = 0.3; Thi = 0.5; di = 0.1; ai = 1.0.

In the non-switching and switching selection term of model A, the role played

by environmental productivity (conveyed here by the ratio of the prey carrying

capacities K1 and K2) in the determination of the food web dynamics was investigated as

follows. The total carrying capacity was held constant, e.g. K1 + K2 = Ktotal = 8.0 and the

ratio K1/K2 varied obeying 1 ≤ K1/K2 ≤ 5. For each ratio value, a bifurcation diagram for

each species as a function of the selection parameter p was built (following the same

procedure as regards time steps). Then, the range of p associated with stable predator

dynamics was plotted as a function of the carrying capacity ratio. The graphical results

for both models are shown in figure 6.

For model NSW – A, figure 6A depicts a decreasing predator stability domain

with increasing carrying capacity ratio. On the other hand, for model SW – A, figure 6B

depicts an increasing predator stability domain with increasing carrying capacity ratio. It

is important to remark that the former is in accordance with the paradox of enrichment,

while the latter is at variance with the same paradox. This result clearly reiterates the

influence of the preference structure modeling on the ultimate food web dynamics.

Figure 6. Stability domains of the predator population dynamics as a function of carrying capacity ratio and the preference term in: (A) model (NSW - A); (B) model (SW - A). Parameter values are the same as in figure 2.

3.4 Discussion

Some considerable effort has been expended in theoretical work to analyze

issues such as palatability and enrichment (Genkai-Kato and Yamamura, 1999), weak

trophic interactions and food web stability (McCann et al., 1998), trophic flow across

habitats (Huxel and McCann, 1998), food web interactions and energetic flows (Teng

and McCann, 2004). A common feature of these works is the prey preference structure

which, according to the cited references, can be modeled in the context of non-

switching predators (sensu Faria et al., 2007) or in the context of switching predators

(sensu Post et al., 2000). In this work a model of two non interacting prey with a shared

predator (NSW – A; SW – A) and a model including a specialist predator (NSW – B;

SW – B) were analyzed under the effect of these two structures of prey preference. It

should be emphasized that this analysis was concerned with illustrating possible

outcomes, rather than providing an exhaustive study of conditions required for all

possible outcomes. Moreover, a complete investigation of the full range of biologically

plausible parameters over the full range of possible initial conditions in models of three

or more species is almost impossible to be carried out (Abrams and Roth, 1994a, b).

Within this context, simulations evidenced a marked influence of the selection

structure on the dynamics of these models. The non-switching preference structure

suggests that depending on the degree of the selection term p, it can stabilize the

consumer-resource dynamics, whereas the switching preference structure generates

instability (e.g. limit cycles) for all degrees of the selection term p. On the other hand,

the specialist inclusion promotes complex behavior (e.g. chaos) to the system with non-

switching preference term; meanwhile it promotes limit cycles to the switching

preference system, eliminating the complex behavior. It is interesting to observe that -

for same values of parameters to the models - both switching models, SW – A and SW

– B, result in limit cycles through all values of p, but non-switching models may

produce different dynamics (i.e. stable state, limit cycles and complex behavior).

As to environmental productivity gradient, the analysis of stability domain for

both frameworks of model (A) (fig. 6) points out that the selection structures employed

here may change completely the food web dynamics. Non-switching structure (NSW –

A) can increase the predator instability domain when this gradient increases, whereas

switching structure (SW – A) rather decreases it. With regard to the paradox of

enrichment proposed by Rosenzweig (1971), which suggests that prey-predator

dynamics eventually becomes unstable when productivity is augmented, these results

indicate that the preference structure may be crucial in the determination of the

consumer-resource dynamics, reverting even the direction stated by the paradox.

Another context to which the analyzed models may apply concerns Integrated

Pest Management (IPM – Stern et al., 1959) - a pest control strategy consisting of

simultaneous pesticide spreading and biocontrol agents. Considering a pesticide induced

mortality rate proportional to the population level of each species, this mortality agent

could be cast as a reduced intrinsic growth rate ri, a reduced carrying capacity Ki for

both prey, and an increased per capita mortality rate di for the predator(s). Hence, the

dynamical results presented in this work could serve as a guideline to assess the

efficiency of Integrated Pest Management in a particular food web.

It is very likely that this dynamical variety may extend to other complex food

webs. More importantly, the analysis presented here emphasizes that modeling

preference structure should be taken with care with respect to the analysis of its

influence on the relation among species interactions, stability and diversity in food

webs.

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