Ecology 302: Lecture VII. Species Interactions. (Gotelli, Chapters 6; Ricklefs, Chapter 14-15) MacArthur s warblers. Variation in feeding behavior allows morphologically similar species of the genus Dendroica to coexist in eastern coniferous forests. 1
Key Points. Types of pair-wise interactions and examples. Lynx-hare cycle. o Embedded in a complex food web. o Supplemental hare feeding / hare predator exclusion experiments suggest that both predation (lynx eats hare) and hare food supply essential components of the cycle. o Validated by three-level food chain model: Lynx-harevegetation. Lotka-Volterra (L-V) predator-prey model. o Prey grow exponentially absent predation o Predators harvest prey in proportion to their abundance. o Isocline analysis oscillatory solutions. o Global analysis infinite number of neutrally stable periodic orbits. In the laboratory, one or both species of a predator-prey pair go extinct unless special measures taken. o Increasing microcosm size. 2
o Reducing prey food supply (medium impoverishment) o Reducing contact rate between predator and prey o Spatially heterogeneous environment. L-V model biologically unrealistic. Consumer-resource equations cap victim density, i.e., absent predation, victims assumed to grow logistically. o System goes to stable equilibrium with both species present or o Predators die out and o Depends on whether or not predator numbers can increase when. Rosenzweig-MacArthur equations add predator satiation. o Two species equilibrium can be stable or unstable o If unstable, dynamics a single stable limit cycle. o Depends on degree of predator-proficiency. o Paradox of enrichment. 3
I. Pair-wise Interactions. A. Competition (-/-). 1. Each species reduces the growth rate of the other. 2. Can be exploitative or via interference. 3. Resources can be renewable (seeds, in-sects, etc.) or not (space, nest cavities). B. Predator-Prey (+/-) 1. One species benefits; the other suffers. 2. Includes pathogen-host, parasite-host interactions. C. Mutualism (+/+) 1. Can be obligate e.g., a. Yucca-Yucca moth. b. Acacia-ant. 2. Or not. Figure 1. Examples of predatorprey, mutualistic and commensal interactions. 4
D. Commensalism (+/0) 1. One species benefits from, but does not affect the other. 2. E.g. Cattle egrets / buffalo. E. Amensalism (-/0) 1. One species harmed by, but does not affect, the other. 2. Understory (short) trees are shaded out by, but do not affect, the growth of canopy (tall) species. Figure 2. Above. Competition for space by two species of barnacles. Next page. Ant-acacia mutualism. 5
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F. Nature of interaction can change with circumstamces. G. Ricklefs gives example of saguraro-nurse plant interaction: 1. Saguaro protected by nurse plants when young (+/0) 2. Ungrateful older saguaro unaffected by, but interfereres with, nurse plant (0/-). Or the two plants compete with each other for water (-/-). 7
II. Lynx-Hare Cycle. A. Pairwise interactions embedded in larger networks. 1. Sometimes, behavior of the whole understandable in terms of a limited number of key interactions. 2. Example: Boreal forest foodweb. a. Arrows indicate flow of energy who eats whom. b. Much can be understood in terms of three variables: lynx, hare and vegetation. Figure 3. The lynx-hare interaction is embedded in a larger network. 8
B. Background. 1. Ten-year cycle since late 1700s. 2. Lynx eat hare (principally); hares eat vegetation. 3. Long-standing dispute as to nature of the oscilation. C. Field experiments both important. D. Three treatments: 1. Hares given supplemental food. 2. Terrestrial predators excluded by fences through which hares could pass. 3. Food + Exclusion Figure 4. Top. Lynx pursues hare. Bottom. Time series. 9
E. Results: 1. Supplemental food increased hare densities, but didn t delay population crash. 2. Excluding terrestrial (but not avian) predators had negligible effect. 3. Combined treatment increased hare densities and postponed crash. F. Three species model. Figure 5. Effect of supplemental feeding and partial predator exclusion on snowshoe hare demography. Top. Data of Krebs and associates. Bottom. Output of three species model 1. Lynx, hare, vegetation + seasonality. 2. Parameterized from known biology. 3. Qualitatively replicates experimental findings. 10
III. Lotka-Volterra Predator-Prey Dynamics. A. Equations. (1) 1. P predators; V victims; 2. baby predators per victim consumed; k kill rate pd-sec ; d predator per capita death rate sec. 3. r victim per capita rate of increase absent predation sec ; B. Equilibrium: ; (2) 11
IV. Isocline Analysis. A. Plot zero-growth isoclines: 0: 0: (3) B. Note whether each species increases or decreases in different regions of plane. 0 0 (4) Figure 4. Predator ( ) and victim ( ) zerogrowth isoclines. Arrows indicate changing densities of the two species. There are two equilibria,, /, and the origin, which is a saddle. C. Draw arrows in P and V directions; compute resultants. D. We can conclude that the system can oscillate, but not whether the oscillations die out, grow or tend to one or socalled limit cycles. 12
V. Global Dynamics. A. One can prove (but we will not) that 1. There are an infinite number of oscillations that neither grow nor decay. 2. Amplitude and period of the oscillations depends on the initial values of the two species. Figure 5. Lotka-Volterra dynamics. left. Isocline analysis indicates the potential for oscillatory behavior. Right. Solutions to Eqs 1. Changing the constants distorts the solution curves, but does not affect the qualitative picture an infinite number of neutrally stable cycles. 13
VI. In the Laboratory. A. Early experiments by Gause and others 1. Confirmed oscillatory nature of Pd-Py dynamics, but 2. Absent immigration, one or both species invariably go extinct. B. Subsequently determined that oscillations stabilized by 1. Increasing size of the microcosm no stochastic extinction (Luckenbill) 2. Medium impoverishment (Luckenbill) as opposed to enrichment which destabilizes (see Gotelli, pp. 140 ff.) 3. Mobility reduction (both species) reduces contact rate by making the medium more viscous (Luckinbill). 4. Spatial heterogeneity Huffaker s mite expts (see Rickleffs, p. 310 ff.) Figure 5. Left. Didinium eats Paramecium. Right. When introduced to a population of Paramecium at carrying capacity, Didinium exterminates its prey and then dies out. From Gause (1934) 14
VII. Beyond Lotka-Volterra. A. Unrealistic L-V assumptions. 1. No limit to numbers of prey absent predation. 2. Per predator harvest rate. Requires a. infinite predator hunting, killing skills; b. Infinitely distensible predator stomachs 3. No stochastic extinction. So many mice; so little time. B. Other unrealistic assumptions. a. Only two species. b. Populations homogeneous all individuals the same. c. Space doesn t matter well stirred assumption. d. Phenotypes fixed no behavioral, developmental, evolutionary responses to changing numbers of both species. 15
Figure 6. Potpourri of predator-prey dynamics. Representative trajectories superimposed on zero-growth isoclines and a. Lotka-Volterra model (Equations 1) reproduces the back and forth motion of a frictionless pendulum. b. With the addition of an upper bound to victim density, infinite numbers of periodic solutions give way to a single stable equilibrium. c. The addition of predator satiation restores the possibility of oscillatory behavior. d. Providing for extinction when victim densities drop below a critical threshold converts the system into an ecological analog of a nerve cell stimulate and fire. 16
(5) C. Consumer-Resource Equations cap victim population. 1. Equations. 1 2. If there is an interior equilibrium (Figure 4b) at ^ 1 (6) ^ 3. Two scenarios: a. :, 0, b. :,, ;, 0, is a saddle. 17
Figure 7. Isoclines (top) and dynamics (bottom) in the case of victim limitation in the absence of predators (Eqs 5). Left.. Right.. Note that in this and the preceding (L-V) case, the origin in as saddle. 18
D. Nonlinear Functional Response (FR) cap predator consumption. 1. Replace in LV and CR with, where (7) and χ is called the halfsaturation constant. 2. is called the predator functional response (FR). 3. Note that lim 1 4. Type II FR ( 1). increases with V at a diminishing rate. Figure 8. Realizations of Eq 7 (functional response) for different values of n (top) and χ (bottom). With n = 1, FR is equivalent to Michaelis-Menten kinetics. 5. Type III FR. ( 1. is sigmoidal. 19
E. Rosenzweig-MacArthur (RM) model: Prey carrying capacity + type II FR. 1. Equations. (8) 2. Equilibria. 1, 0,0 ;, 0,,, 1 (9) 3. Isoclines. 0: 1 (10) 0: 20
4. Victim isocline dome-shaped; peaks at 2 11 5. Three scenarios: a. :, 0, b. :,. c. :, a limit cycle. Figure 9. Predator-prey dynamics with victim carrying capacity and type II FR. Left. J. The interior equilibrium is stable. Right. Solutions tend to a stable limit cycle. 21
6. Paradox of enrichment: Increasing victim carrying capacity. a. Equilibrium density of predators increased. b. Equilibrium density of victims unaffected. c. Can destabilize otherwise stable interaction, Figure 10. Enriching a food chain at its base can destabilize a predator-prey interaction by shifting the peak in the victim isocline to the right of the predator isocline. i.e., stable equilibrium becomes unstable; system winds out to a stable limit cycle. d. Consequence of shifting χ /2 (victim density at which victim isocline peaks) to the right. 22
Table 1. Dynamics of Predator-Prey Models. Model, (0,0) (0,K) POs Neut. Number LV Saddle Stable Neut. Stable Saddle or CR Stable Saddle RM Stable Unstable Saddle stable Saddle or stable Limit cycle 23