The role of metabolic trade-offs in the establishment of biodiversity Stochastic Models in Ecology and Evolutionary Biology - Venice Leonardo Pacciani 5 th April 2018
Introduction Open questions 1 of 18
Introduction Open questions 1 What is the relationship between an ecosystem’s biodiversity and its stability? → May’s stability criterion 1 of 18
Introduction Open questions 1 What is the relationship between an ecosystem’s biodiversity and its stability? → May’s stability criterion 2 How many species can compete for the same resources? → Competitive exclusion principle 1 of 18
Introduction May’s stability criterion 2 of 18
Introduction May’s stability criterion The first theoretical criterion for ecosystem stability was introduced by May in 1972 1 . 1 Robert May. “Will a Large Complex System be Stable?” In: Nature 238 (1972). 2 of 18
Introduction May’s stability criterion The first theoretical criterion for ecosystem stability was introduced by May in 1972 1 . Main result Building a very simple model of ecosystem governed only by stochasticity and characterized by its biodiversity, i.e. the number m of species present, the system will be stable only if √ Σ mC < d (1) with Σ, C and d parameters of the model that describe the inter-specific interactions (contained in the community matrix ). 1 Robert May. “Will a Large Complex System be Stable?” In: Nature 238 (1972). 2 of 18
Introduction May’s stability criterion The first theoretical criterion for ecosystem stability was introduced by May in 1972 1 . Main result Building a very simple model of ecosystem governed only by stochasticity and characterized by its biodiversity, i.e. the number m of species present, the system will be stable only if √ Σ mC < d (1) with Σ, C and d parameters of the model that describe the inter-specific interactions (contained in the community matrix ). Problem Biodiversity brings instability, but observations suggest the opposite! 1 Robert May. “Will a Large Complex System be Stable?” In: Nature 238 (1972). 2 of 18
Introduction May’s stability criterion 3 of 18
Introduction May’s stability criterion A more general formulation 2 It is possible to generalize May’s simple model in order to make it more realistic. In the end the stability criterion can be written as � √ � m V (1 + ρ ) − E , ( m − 1) E max < d , (2) where again V , E , ρ and d are again parameters of the community matrix . 2 Stefano Allesina and Si Tang. “The stability–complexity relationship at age 40: a random matrix perspective”. In: Population Ecology 57.1 (2015). 3 of 18
Introduction Competitive exclusion principle 3 3 G Hardin. “The Competitive Exclusion Principle”. In: Science 131.November (1959), pp. 1292–1297. 4 of 18
Introduction Competitive exclusion principle 3 species S 1 species S 2 resource R 1 . . . . . . . . . species S p resource R p . . . species S m > p 3 G Hardin. “The Competitive Exclusion Principle”. In: Science 131.November (1959), pp. 1292–1297. 4 of 18
Introduction Competitive exclusion principle 3 species S 1 species S 2 resource R 1 . . . . . . . . . species S n ≤ p resource R p . . . species S m > p 3 G Hardin. “The Competitive Exclusion Principle”. In: Science 131.November (1959), pp. 1292–1297. 4 of 18
Introduction Competitive exclusion principle 3 species S 1 species S 2 resource R 1 . . . . . . . . . species S n ≤ p resource R p . . . species S m > p It is violated in many cases: paradox of the plankton . 3 G Hardin. “The Competitive Exclusion Principle”. In: Science 131.November (1959), pp. 1292–1297. 4 of 18
The PTW model 5 of 18
The PTW model Recently, Posfai et al. have proposed a model 4 that exhibits interesting properites. 4 Anna Posfai, Thibaud Taillefumier, and Ned S Wingreen. “Metabolic Trade-Offs Promote Diversity in a Model Ecosystem”. In: Physical Review Letters 118.2 (2017). 5 of 18
The PTW model Recently, Posfai et al. have proposed a model 4 that exhibits interesting properites. � p � � n σ = n σ ˙ α σ i r i ( c i ) − δ (3a) i =1 � m � � c i = s i − ˙ n σ α σ i r i ( c i ) − µ i c i (3b) σ =1 4 Anna Posfai, Thibaud Taillefumier, and Ned S Wingreen. “Metabolic Trade-Offs Promote Diversity in a Model Ecosystem”. In: Physical Review Letters 118.2 (2017). 5 of 18
The PTW model Recently, Posfai et al. have proposed a model 4 that exhibits interesting properites. “metabolic strategies” � p � � n σ = n σ ˙ α σ i r i ( c i ) − δ (3a) uptake of resource i as i =1 a function of its density, � m � ci e.g. ri ( ci ) = � 1+ ci c i = s i − ˙ n σ α σ i r i ( c i ) − µ i c i (3b) σ =1 resource supply 4 Anna Posfai, Thibaud Taillefumier, and Ned S Wingreen. “Metabolic Trade-Offs Promote Diversity in a Model Ecosystem”. In: Physical Review Letters 118.2 (2017). 5 of 18
The PTW model Recently, Posfai et al. have proposed a model 4 that exhibits interesting properites. “metabolic strategies” � p � � n σ = n σ ˙ α σ i r i ( c i ) − δ (3a) uptake of resource i as i =1 a function of its density, � m � ci e.g. ri ( ci ) = � 1+ ci c i = s i − ˙ n σ α σ i r i ( c i ) − µ i c i (3b) σ =1 resource supply Assumptions: 4 Anna Posfai, Thibaud Taillefumier, and Ned S Wingreen. “Metabolic Trade-Offs Promote Diversity in a Model Ecosystem”. In: Physical Review Letters 118.2 (2017). 5 of 18
The PTW model Recently, Posfai et al. have proposed a model 4 that exhibits interesting properites. “metabolic strategies” � p � � n σ = n σ ˙ α σ i r i ( c i ) − δ (3a) uptake of resource i as i =1 a function of its density, � m � ci e.g. ri ( ci ) = � 1+ ci c i = s i − ˙ n σ α σ i r i ( c i ) − µ i c i (3b) σ =1 resource supply Assumptions: 1 µ i = 0 4 Anna Posfai, Thibaud Taillefumier, and Ned S Wingreen. “Metabolic Trade-Offs Promote Diversity in a Model Ecosystem”. In: Physical Review Letters 118.2 (2017). 5 of 18
The PTW model Recently, Posfai et al. have proposed a model 4 that exhibits interesting properites. “metabolic strategies” � p � � n σ = n σ ˙ α σ i r i ( c i ) − δ (3a) uptake of resource i as i =1 a function of its density, � m � ci e.g. ri ( ci ) = � 1+ ci c i = s i − ˙ n σ α σ i r i ( c i ) − µ i c i (3b) σ =1 resource supply Assumptions: 1 µ i = 0 2 ˙ c i = 0, so r i ( c i ) → r i ( � n ) 4 Anna Posfai, Thibaud Taillefumier, and Ned S Wingreen. “Metabolic Trade-Offs Promote Diversity in a Model Ecosystem”. In: Physical Review Letters 118.2 (2017). 5 of 18
The PTW model Recently, Posfai et al. have proposed a model 4 that exhibits interesting properites. “metabolic strategies” � p � � n σ = n σ ˙ α σ i r i ( c i ) − δ (3a) uptake of resource i as i =1 a function of its density, � m � ci e.g. ri ( ci ) = � 1+ ci c i = s i − ˙ n σ α σ i r i ( c i ) − µ i c i (3b) σ =1 resource supply Assumptions: 1 µ i = 0 2 ˙ c i = 0, so r i ( c i ) → r i ( � n ) 3 � p i =1 α σ i = E ∀ σ 4 Anna Posfai, Thibaud Taillefumier, and Ned S Wingreen. “Metabolic Trade-Offs Promote Diversity in a Model Ecosystem”. In: Physical Review Letters 118.2 (2017). 5 of 18
The PTW model Recently, Posfai et al. have proposed a model 4 that exhibits interesting properites. “metabolic strategies” � p � � n σ = n σ ˙ α σ i r i ( c i ) − δ (3a) uptake of resource i as i =1 a function of its density, � m � ci e.g. ri ( ci ) = � 1+ ci c i = s i − ˙ n σ α σ i r i ( c i ) − µ i c i (3b) σ =1 α σ 1 resource supply Assumptions: 1 µ i = 0 2 ˙ c i = 0, so r i ( c i ) → r i ( � n ) 3 � p α σ 3 α σ 2 i =1 α σ i = E ∀ σ 4 Anna Posfai, Thibaud Taillefumier, and Ned S Wingreen. “Metabolic Trade-Offs Promote Diversity in a Model Ecosystem”. In: Physical Review Letters 118.2 (2017). 5 of 18
The PTW model 6 of 18
The PTW model Main result The system will reach a stationary state where an arbitrary number of species can coexist if p m m E � � � n ∗ n ∗ S � s = σ � α σ with σ = 1 , S = s i , (4) σ =1 σ =1 i =1 has a positive solution n ∗ σ > 0. This means that coexistence is possible if � sE / S belongs to the convex hull of the metabolic strategies. 6 of 18
The PTW model Main result The system will reach a stationary state where an arbitrary number of species can coexist if p m m E � � � n ∗ n ∗ S � s = σ � α σ with σ = 1 , S = s i , (4) σ =1 σ =1 i =1 has a positive solution n ∗ σ > 0. This means that coexistence is possible if � sE / S belongs to the convex hull of the metabolic strategies. Important remark The number of coexisting species is arbitrary , so we can also have m > p : the competitive exclusion principle can be violated. 6 of 18
The PTW model 7 of 18
● ● ● ★ ● ● ● ● ● ● ● ★ ● ● ● ● ● ● ● ● ● The PTW model 10 0 10 - 1 10 - 2 10 - 3 10 - 4 10 - 5 10 - 6 0 200 400 600 800 1000 m = 15, p = 3, n σ (0) = 1 / m ∀ σ 7 of 18
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