Hao Wang, Arizona State University John Nagy, Scottsdale Community College Yang Kuang, Arizona State University

1. Top-down Trophic Dynamics produce the 4-year Lemming Cycle and the 10-year Snowshoe Hare Cycle: the maturation delay of predators and the functional response control prey population cycle Hao Wang, Arizona State University John Nagy, Scottsdale Community College Yang Kuang, Arizona State University Olivier Gilg, University of Helsinki, Finland

2. Motivation What are the ecological mechanisms to generate population fluctuations with different periods? Can these mechanisms explain all cyclic populations? Do these mechanisms allow us to understand why some populations are cyclic whereas others are not? Hao Wang - MTBI workshop in ASU

3. Historic Predator-Prey System For the collared lemmings in NE Greenland, predation is the most probable force driving the cyclic dynamics. A predator-lemming model based on the field-estimated predator responses robustly predicting 4-year periodicity in lemming dynamics, in agreement with long-term empirical data. (O. Gilg et al, Science, 2003) Hao Wang - MTBI workshop in ASU

4. Short Gateway Ecological possible mechanisms to generate the lemming cycle are: random effect, migrations/ dispersals, Weather and synchrony, food supply, predation, disease and parasites, and body weight changes. Most ecologists trust that such oscillations are controlled by a trophic mechanism: either an interaction between food supply and lemmings (called bottom-up trophic dynamics), or an interaction between lemmings and predators (called top-down trophic dynamics). Bottom-up trophic dynamics with lemming maturation delay has the 5.5-6 year period under appropriate parameterization. Top-down trophic dynamics with predator maturation delay has the 4 year period under well estimated parameterization in Greenland. Therefore … … Hao Wang - MTBI workshop in ASU

5. Functional Response Test Predation by stoat is modeled with Holling Type III functional response, which was used to incorporate a possible "refuge" for the lemming at very low densities. when lemmings are so dispersed, then they must become very hard to locate for the stoat. Hao Wang - MTBI workshop in ASU

6. Stoat Maturation Delay The stoat maturation delay is about 3 months. The stoat juvenile/maturation death rate is chosen to be the maximum stoat death rate, 4/year. Hao Wang - MTBI workshop in ASU

7. Prey Dependent Death Rate The stoat death rate depends on lemming density This is tested by Olivier Gilg from field. Mathematically this does not make any difference for the behavior of the system, comparing to a constant death rate. Hao Wang - MTBI workshop in ASU

8. Modified Logistic Growth Per capita growth rate for the lemming This modified growth term does not change the period of the lemming cycle, but indeed change the amplitude (raise the low point). Population density x (John D. Reynolds et al, and Richard M. Sibly et al, Science, 2005) Hao Wang - MTBI workshop in ASU

9. Lemming-Stoat Delayed model As a conclusion of previous slides, the system is Hao Wang - MTBI workshop in ASU

10. Empirical Data Fitting Hao Wang - MTBI workshop in ASU

11. Sensitivity Analysis Increasing stoat maturation death rate (d_j) and delay (tau), or decreasing predation rate (eta) and conversion rate/yield constant (xi) can enlarge the period of the population cycle. Holling Type I will inhibit the cycle. Hao Wang - MTBI workshop in ASU

12. Bifurcation to Confirm Hao Wang - MTBI workshop in ASU

13. Compare to Snowshoe Hare The hare-lynx delayed system is similar to the Lemming-stoat delayed system except Holling Type II functional response, constant lynx death rate. Hao Wang - MTBI workshop in ASU

14. Interpretation The lynx maturation delay is 1.5 years, much larger than the stoat maturation delay, 3 months. Predation rate (eta) and conversion rate (xi) are comparable, lynx maturation death rate (d_j) is less than that of stoat (makes the period smaller), therefore the predator maturation delay is the key factor to generate the 10-year hare cycle. Hao Wang - MTBI workshop in ASU

15. Mathematical Results Without modifications, the general delayed prey-predator system has been analyzed by S. Gourley and Y. Kuang (2004). Similar behaviors occur in our modified system. Hao Wang - MTBI workshop in ASU

16. Conclusion • Only with suitable predator maturation delay and suitable functional response of predation, the population cycle occurs. This partially explain why only a few species have population fluctuations nevertheless most others have no well-regulated population fluctuations. • the predator maturation delay almost determines the period of the prey population cycle. Hao Wang - MTBI workshop in ASU

17. Acknowledge Thanks to Yang Kuang John Nagy Olivier Gilg Jiaxu Li all my group members Hao Wang - MTBI workshop in ASU