Food web stability: the interplay among switching predation
Transcript of Food web stability: the interplay among switching predation
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: [email protected] and [email protected]
______________________________________________________________________
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.
4 References
Abrams, P.A. and Roth, J. 1994a. The responses of unstable food chains to enrichment.
Evolutionary Ecology 8:150-171.
________________. 1994b. The effects of enrichment of three-species food chains with
nonlinear functional response. Ecology 75:1118-1130.
Faria, L.D.B., Umbanhowar, J. and McCann, K. 2007. Multiples predators in
biocontrol: stability, pest suppression and nontarget effects. OIKOS – submitted.
Genkai-Kato, M. and Yamamura, N. 1999. Unpalatable prey solves the paradox of
enrichment. Proceeding of Royal Society of London (Series B) 266:1215-1219.
Holt, R.D. 1983. Optimal foraging and the form of predator isocline. The American
Naturalist 122:521-541.
Huxel, G.R. and McCann, K. 1998. Food web stability: the influence of trophic flows
across habitats. The American Naturalist 152: 460-469.
Huxel, G., McCann, K.S. and Polis, G.A. 2002. Effects of partitioning allochthonous
resources on food web stability. Ecological Research 17:419-432.
Matsuda, H., Kawasaki, K., Shigesada, N., Teramoto, E., Ricciardi, L.M. 1986.
Switching effect on the stability of the prey-predator system with three trophic
levels. Journal of Theoretical Biology 122:251-262.
McCann, K.S. and Hastings, A. 1997. Re-evaluating the omnivory-stability relationship
in food webs. Proceeding of Royal Society of London (Series B) 264: 1249-1254.
McCann, K.S., Hastings, A. and Huxel, G.R. 1998. Weak trophic interactions and
balance of nature. Nature 395: 794-798.
Murdoch, W.W. and Oaten, A. 1975. Predation and population stability. Advances in
Ecological Research 9:1-131.
Polis, G.A. 1991. Complex trophic interactions in deserts: an empirical critique of food
web theory. The American Naturalist 138: 123-155.
Post, M.D., Conners, M.E. and Goldberg, D.S. 2000. Prey preference by a top predator
and the stability of linked food chains. Ecology 81: 8-14.
Rosenzweig, M.L. 1971. Paradox of enrichment: destabilization of exploitation
ecosystems in ecological time. Science 171(969):385-387.
Stern, V.M., Smith, R.F., Bosch, V.D.R. and Hagen, K.S. 1959. The integrated control
concept. Hilgardia 29: 81-101.
Strong, D.R. 1992. Are trophic cascades all wet? The redundant differentiation in
trophic architecture of high diversity ecosystems. Ecology 73: 747-754.
Teng, J. and McCann, K.S. 2004. Dynamics of compartmented and reticulate food web
in relation to energetic flows. The American Naturalist 164:85-100.