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Global analysis of a stoichiometric producer–grazer�model with Holling type functional responses

Hao Wang

Department of Mathematical and Statistical Sciences

University of Alberta

Canada

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Water fleas

Phytoplankton

Light

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Low Light

High Light

(Extra)

High Light

Algal

P:C

Algal C

Daphnia

Extreme high light

🡪Daphnia extinction

Algal C

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Fundamental Reason: C supplies energy to cells, N is essential to build proteins, P is essential to build nucleic acids (DNA&RNA), … …

However, most predator-prey (=consumer-resource) models only consider the carbon flow (=biomass or population), which cannot explain this experiment.

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Flowchart of the model if needed

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Assumptions

The total mass of phosphorus in the entire system is fixed, denoted by P. (closed P, open C)
P:C ratio (=cell quota) in algae varies, but it never falls below a minimum q; Daphnia maintains a constant P:C ratio, denoted by θ. (Homeostasis)
All phosphorus in the system is divided into two pools: phosphorus in daphnia and phosphorus in algae (no phosphorus in media).

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LKE Model

Algal population

(measured by carbon)

Daphnia population

(measured by carbon)

compare P:C ratio in algae

with P:C ratio in daphnia

Loladze et al. “Stoichiometry in Producer-Grazer Systems: Linking Energy Flow with Element Cycling”, BMB, Vol. 62, pp 1137-1162 (2000).

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Outline of the mathematical and numerical methods used to solve the model

Mathematical methods: …

Numerical methods: …

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Holling Type I

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The case K<P is similar, skip figures.

Theorem 3-Theorem 13 present all global stability results

of all equilibrium points of the system with Holling type I

functional response.

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Holling Type II

Li, X., Wang, H. and Kuang, Y., 2011. Global analysis of a stoichiometric producer–grazer model with Holling type functional responses. Journal of mathematical biology, 63(5), pp.901-932.

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Now we start to perform global and bifurcation analysis according to

the varying parameter K (representing the light intensity).

semi

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light

(no limit cycle, the one

plotted is only for proof)

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Summary

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Interpretation of the results

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Take home messages

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Future work

How about smaller e in Type II model?

How about Holling Type III functional response?

… …

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Thank you for your attention!