An Investigation of Population Interaction

DynamicsLindsay Kasuga

November 29, 2006

Math 690

Outline

•Review

•General Interaction Model for two species

•Conditional Interactions

•Alpha Function

•Model

•Graphical Stability Analysis

•Biological Example

Simple Population Growth ModelsSimple Population Growth Models

•Exponential Growth

•Logistic Growth

Time

Popu

latio

n Si

ze

•Biological Interpretation

Commensalisms Mutualism

Parasitism Predator-Prey

General Interaction Model for Two Species

where α, β, γ, δ are constant real parameters.

•The linear terms describe the growth or decay rate of the corresponding population in isolation.

•The nonlinear terms model the interaction of the two populations.

•Problem:

•Parameters are constant

•Static classifications may be inadequate as the outcome of the interaction may vary as a result of densities, size or age of individuals, or environmental conditions.

Conditional Interactions•Definition

•Example: epibionts (sponges, algae) and hosts (crabs, insects, algae)

•Hosts

•Costs: mechanical harm as a result of attachment, including motility impairment, lower nutrient and light availability

•Benefits: epibionts provide protection from predators (camouflage, chemical)

•Epibionts

•Benefits: if attached to motile hosts, increased access to nutrients

•Costs: phototropic epibionts may be impaired if attached to negatively photoactic hosts

• The degree of harm (or benefit obtained) depends largely on the proportion of host surface covered by the epibiont.

Ants and Aphids•Aphids: excrete honeydew, which is rich in sugars and amino acids

•Ants: provide protection against natural predators

•Costs and Benefits

•Benefits depends on the relative densities of the two populations

•At low aphid densities, the benefits are high; at high aphid densities, the benefits are neutral or even negative

Environmental Conditions

• Phloem conditions: the higher the quality of the host plant phloem, the higher the quality of the honeydew secreted, the more ants are attracted, the higher the quality of life for the aphid

• However, the lower the quality of the host plant phloem, the lower the quality of the honeydew produced, and the ants may choose to predate on the aphids

What does this mean?•Populations can be at, or transit between, different combinations of stable equilibrium densities, and sometimes between different types of interactions (Hernandez and Barradas, 2003)

•General model was developed for these dynamics by Hernandez, 1998

is the interaction co-efficient, a function of the impact of population density Nj that can take both positive and negative values.

•Summary

Hernandez 1998 presented an overview of the general model (both obligatory and facultative associations).

The Alpha Co-efficient•Models the effect of the interaction between two species

•If positive, then the presence of species 2 has a positive impact on species 1, via the growth rate

•If negative, then the presence of species 2 has a negative impact on species 1, again via the growth rate

•Magnitude of the alpha co-efficient expresses the intensity of this interaction

•Is a function

•(+,+), (+,-), (-,+), (-,-) notation

•Can be viewed as the algebraic sum of two different functions: the benefits to species 1 and the cost to species 1 (as affected by the presence of species 2)

•The alpha function can take different forms, such as quadratic-ratio, parabolic, exponential and linear

Examples of the Alpha Function

Hernandez and Barradas, 2003

Preliminary Graphical Stability Analysis

Nullclines of the model under four the four different alpha functions

Quadratic ratio Parabolic

Exponential Linear

Hernandez and Barradas, 2003

Quadratic Ratio Alpha Function

•The bi and ci measure the sensitivity of the interaction to changes in the partner’s density and the threshold value between a positive and negative interaction.

•Summary: higher bi magnitudes translate into a more efficient impact of species j on species i. However, ci effects are contrasting. An increase is welcome at high Nj ranges, but is entails less positive alpha values a low Nj.

The Model

Graphical Stability Analysis

Main Conclusions

•In addition to the trivial (0,0) solutions, there are two types of equilibrium densities

•Coexistence-both densities are positive internal solutions

•Exclusion-one species survives at its carrying capacity and the other goes extinct. These are the border solutions.

•Eigenvalues are either both complex with negative real parts or both real.

•When both eigenvalues are real, one can be positive, which implies that if a steady state is unstable, it is always a saddle point.

•Internal stable points can be either nodes or foci, whereas stable borders are always nodes

CharacterizationCase 1

•If both Ki < NjC the system presents at least one internal stable equilibrium, and can have another one, or another two, stable-unstable internal pairs. The border solutions are always unstable

Unstable saddle

Stable internal equilibrium

Hernandez and Barradas, 2003 Back

Case 2

•If one Ki > NjC, there is either one stable-unstable internal pair, or none. The border solution where Ki > NjC is stable, the other one unstable.

Unstable saddle

Stable internal equilibrium

Hernandez and Barradas, 2003

Unstable Border solution

Stable Border Solution

Case 3

•If both Ki > NjC, there is not stable coexistence possible. The two border solutions are stable, meaning one species will be driven to extinction.

Unstable saddle

Hernandez and Barradas, 2003

Stable Border equilibrium

Variable Outcomes

b1 increase

Shift from parasitism to mutualism

c1increases

K1 increases b1 increases;

K1 decreases

Shift from parasitism to exclusion of one species Parasitism to mutualism BackHernandez and Barradas, 2003

Characterization of Variations in the Outcomes

•Homeo-environmental variable outcomes: these occur under the same environmental conditions. Population densities show different stable configurations for the same set of parameters. Transitions between stable states are caused by perturbations. After the perturbation, theenvironmental setting is the same, but is in another stable state.

•Multiple equilibria at coexistence and/or exclusion

•Allo-environmental variable outcomes: variations occur for the same two species but under different environmental conditions. Transitions are caused by changes in environmental conditions.

•Increase/decrease in b1

•Increase/decrease in carrying capacity

Graphical example

Graphical example

Bifurcation

Catastrophic Jumps

Major conclusion of Bifurcation Analysis

•Magnitudes of N1* increase with both K1 and b1 values, but may either decrease or

increase with c1. Hernandez and Barradas, 2003

Stable

Unstable

Parameter Chart

Lower b1 magnitude

Two Stable coexistence

One Stable coexistence

Stable exclusion

One Stable coexistence and one exclusion

Non-shaded areas are single solutionsHernandez and Barradas, 2003

Biological Example

•Whelks and Lobsters

•Perturbation removed lobsters from Marcus Island

•In 1988, 1000 lobster were transferred to Marcus from Malagas, and were gone within a week

Malagas Island Marcus Island

4 km

More Bifurcation

•Bifurcation diagram showing how changes can cause catastrophic jumps from one stable equilibrium to another Hernandez and Barradas, 2003

Removal of predators (lobsters)

Exclusion of lobster

For lobsters to re-invade, K1would have to decrease

Major Conclusions

•Biological association between two species involves both costs and benefits for each partner population

•Ecological conditions determine the range of possible outcomes of a conditional interaction via intrinsic (alpha function) and extrinsic factors (carrying capacities)

•In facultative variable associations, poor extrinsic environmental conditions lead to a stable coexistence of the population (persistence of the association), otherwise, the exclusion of one species in the likely outcome.

•The dynamics of populations with variable interactions are subject to the typical behavior of systems with catastrophic regimes. Small parameter changes might make the system suffer catastrophic jumps between stable branches

•Biologically speaking, small changes in predator abundance might be responsible for huge changes in the nature of an association between partner species.

•The particular history of an interaction is determinant-the ultimate outcome depends on the initial density conditions and on the specific following environmental perturbations.

•Large carrying capacities suggest higher equilibrium densities, high bi values promote positive outcomes with the partner species, and the effect of ci varies with the relative population densities.

References

Hernandez, Maria-Josefine and Ignacio Barradas. 2003. “Variation in the outcome of population interactions: bifurcations and catastrophes.” J. Math.Biol. 46: 571-594.

Hernandez, M.J. 1998. “Dynamics of transitions between population interactions: a nonlinear α-function defined.” Proc. R. Soc. Lond B. 265: 1433-1440.

De Vries, Gerda, Thomas Hillen, Mark Lews, Johannes Müller and BirgittSchönfisch. A Course in Mathematical Biology. Society for the Industrial and Applied Mathematics. 2006.