1. Invasion by exotic species A possible mechanism that allows competitive coexistence between native and exotic plants. Augustina di Virgilio Ewaldo L. de O. Júnior João Pinheiro Neto Luiz H. de Almeida Melina O. Melito Pedro G. A. Alcântara II Southern-Summer School on Mathematical Biology

2. Invasions Native bumblebee Exotic bumblebee Native plants Resources consumption Disease transmission - - + “Steals” nectar II Southern-Summer School on Mathematical Biology

3. Relevance Worldwide phenomena. Invasion can have strong effects on the environment. Diversity of species could be at risk. Conservation polices have to take this into account. II Southern-Summer School on Mathematical Biology

4. Competition A basic competition dynamics should eventually force the elimination of the weaker competitor (Competitive Exclusion Principle). However, there is no evidence to suggest that this is a common occurrence. (Lonsdale 1999; Stohlgren et al. 1999) The species forge a kind of coexistence. II Southern-Summer School on Mathematical Biology

5. So how can there be coexistence? There must be mechanisms regulating the interactions. What could they be? Predator, niche, space, delay for predators to attack (enemy release), or many other possibilities. It could even be that the timescale in which the elimination happens is just too large for we to observe II Southern-Summer School on Mathematical Biology

6. Predator hypothesis Could predators act as a mechanism promoting equilibrium? Two competitive preys one predator. Trade-off: competitive ability X susceptibility to predation? II Southern-Summer School on Mathematical Biology

7. Study system Estuarial plant communities in New England Similar native and exotic plants Herbivory by insects Native species Exotic species (Heard &Sax, 2012) II Southern-Summer School on Mathematical Biology

8. Dynamics – Model 1 Herbivores Native plants Exotic plants II Southern-Summer School on Mathematical Biology

9. Assumptions Natives and Exotics - different growth rates Herbivore rates are different for exotics and natives Competitive strength is not symmetrical Capture rate (  ), conversion rate (  ), and the parameter D are the same for both species II Southern-Summer School on Mathematical Biology

10. First model II Southern-Summer School on Mathematical Biology

11. No predators Coexistence No exotics II Southern-Summer School on Mathematical Biology

12. Population size NativesPredatorsExotics II Southern-Summer School on Mathematical Biology

13. Dynamics – Model 2 Herbivores Native seedlings Exotic seedlings Native adults Exotic adults II Southern-Summer School on Mathematical Biology

14. Second model II Southern-Summer School on Mathematical Biology

15. No predators No exotics No natives Coexistence II Southern-Summer School on Mathematical Biology

16. Robustness of the models II Southern-Summer School on Mathematical Biology

17. Comparison between models II Southern-Summer School on Mathematical Biology

18. General conclusions Both models fit the observations. Predator dynamics could act as a mechanism to promote coexistence between competitors. A basic trade off in adaptability and susceptibility to predators could explain coexistence without loss of biodiversity. We must remember they may not be the only mechanism at work. II Southern-Summer School on Mathematical Biology

19. References: [1] Heard, M.J. and Sax, D.F., Coexistence between native and exotic species is facilitated by asymmetries in competitive ability and susceptibility to herbivores. Ecology Letters 16 (2013) 206. [2] Adler,P.B. et alli, Coexistence of perennial plants: an embarrassment of niches. Ecology Letters 13 (2010) 1019. [3]Keane, R.M. and Crawley, M.J. Exotic plant invasions and the enemy release hypothesis. Trends in Ecology and Evolution 17 (2002) 164. [4] Davis, M.A. et alli. Don't judge species on their origins. Nature 474 (2011) 153. [5] Stromberg, J.C. et alli. Changing Perceptions of Change: The Role of Scientists in Tamarix and River Management. Restoration Ecology 17 (2009) 177 Images from: http://ian.umces.edu/ Augustina di Virgilio - Argentina Special thanks to group 6 for their work on preferences, it was very useful. II Southern-Summer School on Mathematical Biology

20. Parameters – Model 1 Natives growth rate Exotics growth rate Natives carrying capacity Exotics carrying capacity Competition coefficient Feeding efficiency Natives herbivore rate Exotics herbivore rate Saciation coefficient Predators mortality rate Conversion coefficient II Southern-Summer School on Mathematical Biology

21. Parameters range – Model 1 Model parameterValuesRangeReferences Natives growth rate (rn) 0.1 Wilson et al. 1991, Marañon and Grubb 1993, Hoffmann and poortman 2002 Exotics growth rate (re) 0.2r2 < 0.3 Marañon and Grubb 1993, Hoffmann and poortman 2002 Natives carrying capacity (Kn)100.0 Exotics carrying capacity (Ke)80.0K2<100 Competition coefficient (beta) 0.013beta < 0.016 Levins and culver 1971, Hulbert 1978 Feeding efficiency (theta) 0.09theta < 7 Wilson et al. 1991 Saciation coefficient (D)20.06 < D < 45 Ben-Shahar and robinson 2001 Natives herbivory rate alfa n 0.30 Pacala y Tilman 1994 Exotics herbivory rate alfa e0.7 alfa 2 > 0.33 Pacala y Tilman 1994 Predators mortality rate (mu) 0.030.013<mu<0.044 Anderson and ray 1980, Wilson et al. 1991, Chapman et al. 1998 Conversion coefficient (gamma) 0.06 Wilson et al. 1991 II Southern-Summer School on Mathematical Biology

22. Parameters range – Model 2 Model parameterValuesRangeReferences Natives growth rate (rn) 0.1 Wilson et al. 1991, Marañon and Grubb 1993, Hoffmann and poortman 2002 Exotics growth rate (re) 0.2Marañon and Grubb 1993, Hoffmann and poortman 2002 Carrying capacity (Kn) 100.0 Prop. seedling to adults (G) 0.01G < 0.3 Natives herbivore rate alpha n 0.90.01 < alfa 1< 5Pacala y Tilman 1994 Competition coefficient (beta) 0.01beta < 0.3Levins and culver 1971, Hulbert 1978 Natives mortality rate (mu) 0.03 Exotics mortality rate (mu) 0.070.05 <mu< 0.15 Seedlings to adults ratio for support 1.1< 1.1 Predators mortality rate (mu) 0.028 Conversion coefficient (gamma) 0.20.15 <gamma < 0.5Wilson et al. 1991 Feeding efficiency (theta) 0.090.01<theta<7Wilson et al. 1991 Saciation coefficient (D) 100.0Ben-Shahar and robinson 2001 II Southern-Summer School on Mathematical Biology

23. Experimental observations Results (Heard & Sax, 2012): II Southern-Summer School on Mathematical Biology