Do They Stay or Do They Go Now?

1. Do They Stay or Do They Go Now? Habitat selection of aquatic beetles and its impact on spatial distribution By Drew Hanson & Justin Marleau May 11th, 2007 PIMS Mathematical Biology Summer Workshop

2. Introduction • The most utilized research programme for the explanation of species distributions and abundances in community ecology is random dispersal followed by non-random, site-specific mortality (Binckley & Resetarits 2005). • While other research programmes have been developed involving different mechanisms, such as neutral theory (see Hubbell 2001), few have incorporated important behavioural mechanisms which can possibly alter species distributions and abundances.

3. Introduction • One such mechanism is habitat selection, which involves organisms dispersing and colonizing patches with the highest expected fitness. • Our goal is to model this mechanism and determine if there are significant differences between the emergent distribution and abundance of organisms predicted by random dispersal and non-random site-specific mortality.

4. The Organism: Beetle • Tropisternus lateralis, a common predaceous diving water beetle. • Have an initial dispersal flight after hatching that can be over a half-mile (800 metres) in distance (Milliger & Schlicht 1968, Wallace & Anderson 1996).

5. The Habitat: Ponds • T. lateralis occurs naturally in small ponds (like rice paddies, temporary woodland ponds) and is considered to be an important species in aquatic systems as a predator and a prey (Resetarits 2001).

6. Proposed Mechanisms of Habitat Selection • Pond Size: Beetles are attracted to shiny surfaces, so larger ponds would be more likely to be seen by beetles flying overhead and therefore be colonized by beetles after their dispersal flight (Wallace & Anderson 1996). • Presence of predator: Certain species of beetles are hypothesized to be able to detect chemically the presence of predators once in the pond, making them more likely to leave ponds with predators (Binckley & Resetarits 2005).

7. Other Relevant Biological Information • Beetles in ponds containing predators are less active than in ponds containing no predators (Resetarits 2001). • Ponds can become rapidly saturated with beetles as they preferentially stay on the side of ponds, causing the beetles to leave and colonize other ponds once a threshold level of abundance is reached (Binckley & Resetarits 2005).

8. How does one model the system? • As the beetles are undergoing long-distance dispersal, we will be using a gravity model (Bossenbroek et al. 2001). • A gravity model, which is based on Newton’s law of gravity, assumes that individuals will be attracted to large areas, i.e. large ponds in our model. • Other components can affect the “attractiveness” of lakes, such as the nutrient composition and the presence of predators.

9. The Breakdown: Outline of Talk • 1. Creating an appropriate gravity model and qualitatively analyse it, without regard to presence of predators. • 2. Introducing the predators, but not introducing predation, in order to compare with results of Binckley & Resetarits (2005). • 3. Introduce predation and multiple generations for a “realistic” model.

10. Part I: The Set-Up • Initial Model Assumptions: • All ponds have same concentrations of nutrients for beetles. • There is no mortality of beetles over the time-frame considered. • Each pond, depending on its size, has a carrying capacity of beetles. Beetles will not leave a pond if it is under carrying capacity. • The model is governed by deterministic equations.

11. Part I: The Set-up • Hundred randomly distributed ponds of variable size. Original population starts at (0,0) at time 0.

12. Part I: Parameters for initial deterministic model • T = Total number of beetles in the system of ponds • Tj= Total number of beetles in pond j • Kj = Carrying capacity of pond j • Ai = Balancing coefficient that ensures all beetles leaving pond i arrive to some pond in the system • cij = distance from pond i to pond j • N0j = Number of beetles arriving from outside the pond system that arrive to pond j during their dispersal flight • Wj= Attractiveness of pond j • Mi = Number of beetles leaving pond i • a = Distance coefficient • P = Total number of ponds in our system

13. 1 2 Part I: The deterministic equations

14. Part I: Parameter values • Wj = area of pond j in m2 • Kj= (1/0.435m2)*60*(area of pond j) (Note: The (1/0.435m2)*60 term is derived from Binckley & Resetarits (2005)) • a= 1.9 • T = 15 000

15. Part I Results: 15000 at (0,0) at t=0

16. Part I: If we start elsewhere?

17. Part I: If we have more beetles?

18. Part II: Adding greater realism • Distance and size of ponds are not the only factors determining the abundance and distribution of beetles, the presence of fish are also very important. • Fish act in two ways: they reduce the attractiveness of ponds by reducing the perceived carrying capacity of ponds and they consume beetles.

19. Part II: Beetle Eating Machine • Enneacanthus obesus: blue-stripped sunfish

20. Part II: The Effects of the Blue-Stripped Sunfish • Can reduce its prey population by 70% in a single day at an intermediate size (3.75g) (Chalcraft & Resetarits 2004). • Mere presence reduces the attractiveness of ponds (carrying capacity) by 80% to beetles and greatly decreases the activity of beetles remaining in the ponds (Resetarits 2001, Binckley & Resetarits 2005).

21. Part II: Investigating non-lethal impact of predators • In order to see if our model could properly model the non-lethal effects observed by Resetarits and associates, we re-created one of their experimental set-ups within our model framework and we assumed that the carrying capacity of ponds was the substantial difference between ponds (as was the case with Binckley & Resetarits 2005).

22. Part II: Visual Representation of Binckley & Resetarits (2005)

23. Part II : Results

24. Part III: The “generalized” model • In this section, we allow for predation, multiple generations of beetles and the carrying capacity is lowered by the presence of fish. • As beetles are univoltine (one generation alive at a time), we assume that beetles lay eggs (10 eggs per adult) near the end of the year, and expire before the next generation is hatched. (Zola et al. 1980). • We also assume that newly-hatched beetles all leave their original pond to colonize new ponds, but do not leave the pond system.

25. Part III: Changes to Equations • Tj= Tj – min(0.7*Tj *number of fish in pond j, 70*number of fish in pond j) • The above equation takes into account the voracious appetite of the predators in modifying the beetle populations.

26. Part III: Result

27. Part III: Differences between detecting and non-detecting beetles

28. Part III: Differences between detecting and non-detecting beetles

29. Part III: Differences between detecting and non-detecting beetles • After one hundred random samplings (from different starting points), we discovered: • Mean of detecting beetles: 5345 • Standard Deviation: 707 • Mean of non-detecting beetles: 4979 • Standard Deviation: 759

30. Part III: Differences between detecting and non-detecting beetles • These differences are highly dependent on when the predation, the number of predators present, the location of the predators and the birth rates of the beetles. Greater analysis is needed before any firm conclusions can be drawn.

31. Part III: Pros and Cons PROS • The beetle population, regardless of initial conditions, goes to a global steady state over some number of years. • The model is straightforward, and seems to accurately fit limited experimental data that exists. CONS • The assumption of habitat selection reduces the overall capacity of the pond system. This leads to a increased population density of selective beetles rather than an increased overall population. • Birth rates are constant among all beetles, while realistically, beetles who are living in a pond with predators have much lower fitness.

32. Summary • It is possible and useful to model dispersal flights of beetles as a simple gravity model. • The gravity model with changes in carrying capacity due to the presence of predators is capable of generating results that are similar to those of experimental studies. • There can be global steady-states independent of initial conditions and beetle detection strategy if multiple generations are considered and the birth rates are large enough. • The actual dynamics of multiple generations is highly dependent on parameter values and require greater study.

33. Acknowledgements • We would like to thank Mark and Caroline for their insightful advice in the formulation of our project as well as everyone participating and organizing this workshop for creating such a great atmosphere.

34. References • Binckley, C.A. & Resetarits, W. 2005 Habitat selection determines abundance, richness and species composition of beetles in aquatic communities. Biol. Lett. 1, 370-374. • Resetarits, W. 2001 Colonization under threat of predation: avoidance of fish by an aquatic beetle, Tropisternus lateralis (Coleoptera: Hydorphilidae). Oecologia129, 155-160. • Wallace, J.B. & Anderson, N.H. 1996 Habitat, life history and behavioral adaptations of aquatic insects. In An introduction to the aquatic insects of North America (ed. R.W. Merritt & K.W. Cummins), pp. 41-73. Dubuque, Kendall/Hunt.

35. References • Milliger, L.E. & Schlicht, H.E. 1968 Passive dispersal of viable algae and protozoa by an aquatic beetle. Trans. Amer. Microsc. Soc.87, 443-448. • Hubbell, S.P. 2001 Unified Theory of Biodiversity and Biogeography. Princeton. • Bossenbroek, J.M., Clifford, E.K. & Nekola, J.C. 2001 Prediction of long-distance dispersal using gravity models: zebra mussel invasion of inland lakes. Ecological Applications, 11, 1778-1788.

36. References • Zalom, F.G., Grigarick, A.A. & Way, M.O. 1980 Habits and relative population densities of some hydrophilids in California rice fields. Hydrobiologia, 75, 195-200. • Chalcraft, D.R. & Resetarits, W.J. 2004 Metabolic rate models and the substitutability of predator populations. Journal of Animal Biology, 73, 323-332.