1. Dynamical System of a Two Dimensional Stoichiometric Discrete Producer-Grazer Model : Chaotic, Extinction and Noise Effects Yun Kang Work with Professor Yang Kuang and Professor Ying-chen Lai, Supported by Professor Carlos Castillo-Chavez (MTBI) and Professor Tom Banks (SAMSI)

2. Outline of Today’s Talk  Introduce LKE – model, and its corresponding discrete case;  Mathematical Analysis: bifurcation study  Biological Meaning of Bifurcation Diagram;  Chaotic behavior and Extinction of grazer;  Nature of Carry Capacity K and Growth Rate b, and their fluctuation by environments: adding noise  Interesting Phenomenal by adding noise: promote diversity of nature  Conclusion and Future Work

3. Stoichiometry  It refers to patterns of mass balance in chemical conversions of different types of matter, which often have definite compositions

4. most important thing about stoichiometry  we can not combine things in arbitary proportions; e.g., we can’t change the proportion of water and dioxygen produced as a result of making glucose.

5.  Energy flow and Element cycling are two fundamental and unifying principles in ecosystem theory  Using stoichiometric principles, Kuang’s research group construct a two-dimensional Lotka–Volterra type model, we call it LKE- model for short

6. Assumptions of LKE Model  Assumption One: Total mass of phosphorus in the entire system is closed, P ( mg P /l )  Assumption Two: Phosphorus to carbon ratio (P:C) in the plant varies, but it never falls below a minimum q (mg P/mg C); the grazer maintains a constant P:C ratio, denoted by ( mg P/mg C )  Assumption Three: All phosphorus in the system is divided into two pools: phosphorus in the plant and phosphorus in the grazer.

7. Continuous Model  p is the density of plant (in milligrams of carbon per liter, mg C/l);  g is the density of grazer (mg C/l);  b is the intrinsic growth rate of plant (day−1);  d is the specific loss rate of herbivore that includes metabolic losses (respiration) and death (day−1);  e is a constant production efficiency (yield constant);  K is the plant’s constant carrying capacity that depends on some external factors such as light intensity;  f(p) is the herbivore’s ingestion rate, which may be a Holling type II functional response.

8. Biological Meaning of Minimum Functions  K controls energy flow and (P − y)/q is the carrying capacity of the plant determined by phosphorus availability;  e is the grazer’s yield constant, which measures the conversion rate of ingested plant into its own biomass when the plants are P rich ( ); If the plants are P poor ( ), then the conversion rate suffers a reduction.

9. Continuous Case: b=1.2 and b=2.9

10. Discrete Model From Continuous One  Motivation: Data collect from discrete time, e.g., interval for collecting data is a year.  Biological Meaning of Parameters : Modeling the dynamics of populations with non-overlapping generations is based on appropriate modifications of models with overlapping generations.  Choose

12. Mathematical Analysis  We study the local stability of interior equilibrium E*=(x*,y*)

13. Bifurcation Diagram and Its Biological Meaning  For continuous case: K=1.5

14. Bifurcation Diagrams on Parameter b

16. Bifurcation Diagrams on Parameter K

17. Relationship Between K and b:  From these figures, we can see that there is nonlinear relationship between K and b which effect the population of plant and grazer:  For bifurcation of K, increasing the value of b, the diagram of b seems shrink.  For bifurcation of b, increasing the value of K, bifurcation diagram seems move to the left

18. Extinction of Grazer  From bifurcation diagram, we can see that for some range of K and b, grazer goes to extinct. What are the reasons? 

19. Basin Boundary For Extinction

20. Global Stability Conjecture  We know that Discrete Rick Model : x(n+1)=x(n)exp{b(1-x(n)/K)} has global stability for b<2, does our system also has this properties More general, if we have u(n+1)=u(n)exp{f(u(n),0} with global stability, then the following discrete system: x(n+1)=x(n)exp{f(x(n),y(n))+g(x(n),y(n))}, of g(x,y) goes to zero as y tending to zero, in which condition has global stability

21. Nature of K and b  K is carrying capacity of plant, and it is usually limited by the intensity of light and space. Since K is easily affected by the environment, it will not be always a constant ;  b is maximum growth rate of plant, it will fluctuates because of environment changing.

22. Adding Noise  Because of the nature of biological meaning of K and b, it makes perfect sense to think these parameters as a random number.  We let K=K0+ w*N(0,1) b=b0+w*N(0,1)

23. Then Most Interesting thing on parameter K :  Prevent extinction of grazer :

24. Time Windows

25. Scaling  Define the degree of existence : R=average population of graze/ average population of plant Then try different amplititute of noise w, then do the log-log scaling, it follows the scaling law.

26. Future Work  We would like to use “snapshot” method to see how noise effects the population of grazer and producer;  Try to different noise, e.g. color noise, to see how the ‘color’ effect the extinction of the grazer;