II Southern-Summer School on Mathematical Biology
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
Invasions
Resources
consumption
Disease transmission
Native bumblebee
+
-
Native plants
Exotic bumblebee
“Steals” nectar
II Southern-Summer School on Mathematical Biology
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
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
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
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
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
Dynamics – Model 1
Herbivores
Native
plants
Exotic
plants
II Southern-Summer School on Mathematical Biology
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
First model
II Southern-Summer School on Mathematical Biology
No predators
Coexistence
No exotics
II Southern-Summer School on Mathematical Biology
Population size
Natives
Exotics
Predators
Number of Predators at t = 1000
Number of initial predators: 5
Number of Exotics at t = 1000
Number of initial predators: 5
90
80
25
100
30
70
25
80
Exotics at t = 1000
Exotics at t = 1000
20
20
15
15
10
60
60
50
40
40
20
5
10
30
0
80
0
80
100
60
20
5
100
60
80
40
80
40
60
40
20
20
0
Initial exotics
20
0
0
10
60
40
20
0
Initial exotics
Initial natives
Number of Exotics at t = 1000
Number of initial predators: 100
0
Initial natives
Number of Predators at t = 1000
Number of initial predators: 100
90
30
80
35
100
25
30
70
20
20
15
15
10
Exotics at t = 1000
Exotics at t = 1000
80
25
60
60
50
40
40
20
5
30
10
0
80
0
80
100
60
20
100
60
80
40
20
0
0
60
40
20
20
0
Initial natives
80
40
60
40
20
Initial exotics
5
Initial exotics
0
0
Initial natives
10
II Southern-Summer School on Mathematical Biology
Dynamics – Model 2
Herbivores
Native
seedlings
Native
adults
Exotic
seedlings
Exotic
adults
II Southern-Summer School on Mathematical Biology
Second model
II Southern-Summer School on Mathematical Biology
No predators
No exotics
No natives
Coexistence
II Southern-Summer School on Mathematical Biology
Robustness of the models
II Southern-Summer School on Mathematical Biology
Comparison between models
II Southern-Summer School on Mathematical Biology
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
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
Parameters – Model 1
Saciation coefficient
Predators mortality rate
Conversion coefficient
Natives growth rate
Competition coefficient
Exotics growth rate
Feeding efficiency
Natives carrying capacity
Exotics carrying capacity
Natives herbivore rate
Exotics herbivore rate
II Southern-Summer School on Mathematical Biology
Parameters range – Model 1
Model parameter
Values
Natives growth rate (rn)
0.1
Exotics growth rate (re)
0.2
Range
References
Wilson et al. 1991, Marañon and Grubb 1993, Hoffmann and
poortman 2002
r2 < 0.3
Marañon and Grubb 1993, Hoffmann and poortman 2002
Natives carrying capacity (Kn)
100.0
Exotics carrying capacity (Ke)
Competition coefficient
(beta)
80.0
K2<100
0.013
beta < 0.016
Feeding efficiency (theta)
0.09
theta < 7
Wilson et al. 1991
Saciation coefficient (D)
20.0
6 < D < 45
Ben-Shahar and robinson 2001
Natives herbivory rate alfa n
0.30
Exotics herbivory rate alfa e
0.7
alfa 2 > 0.33
Predators mortality rate (mu)
Conversion coefficient
(gamma)
0.03
0.013<mu<0.044
0.06
Levins and culver 1971, Hulbert 1978
Pacala y Tilman 1994
Pacala y Tilman 1994
Anderson and ray 1980, Wilson et al. 1991, Chapman et al.
1998
Wilson et al. 1991
II Southern-Summer School on Mathematical Biology
Parameters range – Model 2
Model parameter
Values
Range
References
Natives growth rate (rn)
0.1
Wilson et al. 1991, Marañon and Grubb 1993, Hoffmann and
poortman 2002
Exotics growth rate (re)
0.2
Marañon and Grubb 1993, Hoffmann and poortman 2002
Carrying capacity (Kn)
100.0
Prop. seedling to adults (G)
0.01
G < 0.3
Natives herbivore rate alpha n
0.9
0.01 < alfa 1< 5
Competition coefficient (beta)
0.01
beta < 0.3
Natives mortality rate (mu)
0.03
Exotics mortality rate (mu)
0.07
0.05 <mu< 0.15
Seedlings to adults ratio for
support
1.1
< 1.1
Predators mortality rate (mu)
Pacala y Tilman 1994
Levins and culver 1971, Hulbert 1978
0.028
Conversion coefficient (gamma)
0.2
0.15 <gamma < 0.5
Wilson et al. 1991
Feeding efficiency (theta)
0.09
0.01<theta<7
Wilson et al. 1991
Saciation coefficient (D)
100.0
Ben-Shahar and robinson 2001
II Southern-Summer School on Mathematical Biology
Experimental observations
• Results (Heard & Sax, 2012):