Models of structured populations in constant and fluctuating - PowerPoint PPT Presentation

Models of structured populations in constant and fluctuating environments Sepideh Mirrahimi CNRS, IMT, Toulouse Toulouse, June 16-19, 2014 1 / 26

Darwinian evolution of a structured population density • We study the Darwinian evolution of a population structured by phenotypical traits, under selection and mutation 2 / 26

Example of evolution: creation of antibiotic resistance of bacteria under drug selection Morbidostat : a selective pressure is applied continuously to the bacterial population. It automatically tunes drug concentration such that a constant growth rate is maintained. Following evolution of bacterial antibiotic resistance in real time, Rosenthal et Elowitz, Nature Genetics 2012 3 / 26

The influence of fluctuating temperature on bacteria Bacterial pathogen Serratia marcesens evolved in fluctuating temperature (daily variation between 24 ◦ C and 38 ◦ C, mean 31 ◦ C), outperforms the strain that evolved in constant environments (31 ◦ C): Figure from: Fluctuation temperature leads to evolution of thermal generalism and preadaptation to novel environments, Ketola et al. 2013 4 / 26

Environment fluctuation can select for generalism ⇒ It can also increase organisms’ ability to invade novel environments. 5 / 26

Environment fluctuation can select for generalism ⇒ It can also increase organisms’ ability to invade novel environments. However, generalism can still be costly in terms of reduced fitness in some ecological contexts: Evolution in fluctuating tempera- ture decreased bacterial virulence in D. melanogoster host: Figure from: Fluctuation temperature leads to evolution of thermal generalism and preadaptation to novel environments, Ketola et al. 2013 5 / 26

A typical model ∂ t n ε − ε ∆ n ε = n ε  ∂ � � ε R x , I ε ,    n ε ( · , t = 0 ) = n 0 ε ( · ) ,  � I ε ( t ) = R d η ( x ) n ε ( x , t ) dx . • x ∈ R d : phenotypical • I ε ( t ) : total consumption trait • R ( x , I ε ) : growth rate • n ε ( x , t ) : density of trait x • ε : a small parameter • η ( x ) : consumption rate 6 / 26

A typical model ∂ t n ε − ε ∆ n ε = n ε  ∂ � � ε R x , I ε ,    n ε ( · , t = 0 ) = n 0 ε ( · ) ,  � I ε ( t ) = R d η ( x ) n ε ( x , t ) dx . • x ∈ R d : phenotypical • I ε ( t ) : total consumption trait • R ( x , I ε ) : growth rate • n ε ( x , t ) : density of trait x • ε : a small parameter • η ( x ) : consumption rate 6 / 26

The typical behavior of the solutions R ( x , I ) = 1 − x 2 A simple typical growth rate: 2 − I Dynamics of the dominant trait 7 / 26

Preliminary properties • − C ≤ ∂ R ∂ I ( x , I ) ≤ − C − 1 < 0 8 / 26

Preliminary properties • − C ≤ ∂ R ∂ I ( x , I ) ≤ − C − 1 < 0 • max x ∈ R d R ( x , I M ) = 0 , 8 / 26

Preliminary properties • − C ≤ ∂ R ∂ I ( x , I ) ≤ − C − 1 < 0 • max x ∈ R d R ( x , I M ) = 0 , • I m ≤ I ε ( 0 ) ≤ I M . 8 / 26

Preliminary properties • − C ≤ ∂ R ∂ I ( x , I ) ≤ − C − 1 < 0 • max x ∈ R d R ( x , I M ) = 0 , • I m ≤ I ε ( 0 ) ≤ I M . = ⇒ I m < I ε ( t ) ≤ I M . 8 / 26

Preliminary properties • − C ≤ ∂ R ∂ I ( x , I ) ≤ − C − 1 < 0 • max x ∈ R d R ( x , I M ) = 0 , • I m ≤ I ε ( 0 ) ≤ I M . = ⇒ I m < I ε ( t ) ≤ I M . Moreover, after extraction of a subsequence, ( I ε ) ε converges a.e. to I ( t ) . 8 / 26

Some notations • n ( x , t ) : weak limit of n ε ( x , t ) as ε vanishes • We expect n to concentrate as Dirac masses � u ε ( x , t ) � • Hopf-Cole transformation: n ε ( x , t ) = exp ε 9 / 26

The Hamilton-Jacobi limit Theorem (Barles, SM, Perthame - 2009) After extraction of a subsequence, u ε converges locally uniformly to a continuous function u, a viscosity solution to ∂ t u = |∇ u | 2 + R ( x , I ( t ))  ∂      x ∈ R d u ( x , t ) = 0 , max    u ( 0 , x ) = u 0 ( x ) .   10 / 26

Convergence to a monomorphic population n ε ( x , t ) − ε → 0 n ( x , t ) , − ⇀ supp n ( x , t ) ⊂ { u ( x , t ) = 0 } ⊂ { R ( x , I ( t )) = 0 } . 11 / 26

Convergence to a monomorphic population n ε ( x , t ) − ε → 0 n ( x , t ) , − ⇀ supp n ( x , t ) ⊂ { u ( x , t ) = 0 } ⊂ { R ( x , I ( t )) = 0 } . In particular, if R : R → R is strictly concave with respect to x , then n ε ( x , t ) − ε → 0 n ( x , t ) = ρ ( t ) δ ( x − x ( t )) , − ⇀ I ( t ) with R ( x ( t ) , I ( t )) = 0 and ρ ( t ) = η ( x ( t )) . 11 / 26

Canonical equation and long time behavior Theorem (Lorz, SM, Perthame - 2013) Under concavity and smoothness assumptions, • n ε ( x , t ) − ε → 0 n ( x , t ) = ρ ( t ) δ ( x − x ( t )) − ⇀ 12 / 26

Canonical equation and long time behavior Theorem (Lorz, SM, Perthame - 2013) Under concavity and smoothness assumptions, • n ε ( x , t ) − ε → 0 n ( x , t ) = ρ ( t ) δ ( x − x ( t )) − ⇀ • x ( t ) and ρ ( t ) are ’smooth’ 12 / 26

Canonical equation and long time behavior Theorem (Lorz, SM, Perthame - 2013) Under concavity and smoothness assumptions, • n ε ( x , t ) − ε → 0 n ( x , t ) = ρ ( t ) δ ( x − x ( t )) − ⇀ • x ( t ) and ρ ( t ) are ’smooth’ • R � � x ( t ) , I ( t ) = 0 , with I ( t ) = η ( x ( t )) ρ ( t ) . 12 / 26

Canonical equation and long time behavior Theorem (Lorz, SM, Perthame - 2013) Under concavity and smoothness assumptions, • n ε ( x , t ) − ε → 0 n ( x , t ) = ρ ( t ) δ ( x − x ( t )) − ⇀ • x ( t ) and ρ ( t ) are ’smooth’ • R � � x ( t ) , I ( t ) = 0 , with I ( t ) = η ( x ( t )) ρ ( t ) . �� − 1 · ∇ x R • ˙ � − D 2 u � � � x ( 0 ) = x 0 x ( t ) = x ( t ) , t x ( t ) , I ( t ) , 12 / 26

Canonical equation and long time behavior Theorem (Lorz, SM, Perthame - 2013) Under concavity and smoothness assumptions, • n ε ( x , t ) − ε → 0 n ( x , t ) = ρ ( t ) δ ( x − x ( t )) − ⇀ • x ( t ) and ρ ( t ) are ’smooth’ • R � � x ( t ) , I ( t ) = 0 , with I ( t ) = η ( x ( t )) ρ ( t ) . �� − 1 · ∇ x R • ˙ � − D 2 u � � � x ( 0 ) = x 0 x ( t ) = x ( t ) , t x ( t ) , I ( t ) , • I ( t ) − t →∞ I M , → x ( t ) − t →∞ x M . with → 0 = R ( x M , I M ) = max x ∈ R d R ( x , I M ) . 12 / 26

x 3 x 2 x M x 1 13 / 26

What if the environment is fluctuating?  ε ∂ x , t ∂ t n ε − ε 2 ∆ n ε = n ε R � � ε , I ε ( t ) ,  n ε ( · , t = 0 ) = n 0 ε ,  � I ε ( t ) := R N η ( x ) n ε ( x , t ) dx , with R 1 -periodic in the second variable . Can we describe the limit as ε → 0 ? 14 / 26

Some assumptions • − 2 K 1 ≤ D 2 x R ( x , s , I ) ≤ − 2 K 2 15 / 26

Some assumptions • − 2 K 1 ≤ D 2 x R ( x , s , I ) ≤ − 2 K 2 • − K 5 ≤ ∂ ∂ I R ( x , s , I ) ≤ − K 6 15 / 26

Some assumptions • − 2 K 1 ≤ D 2 x R ( x , s , I ) ≤ − 2 K 2 • − K 5 ≤ ∂ ∂ I R ( x , s , I ) ≤ − K 6 • max 0 ≤ s ≤ 1 , x ∈ R N R ( x , s , I M ) = 0 15 / 26

Main result Theorem ( SM, Perthame, Souganidis - 2014 ) There exist a fittest trait x and a total population size ρ such that, along subsequences ε → 0 , n ε ( · , t ) − ⇀ ̺ ( t ) δ ( · − x ( t )) weakly in the sense of measures, in L ∞ ( 0 , ∞ ) weak- ⋆, I ε − ⇀ I := ̺η ( x ) and R ( x , t ε, I ε ( t )) − ⇀ R ( x , x ( t )) weakly in t and strongly in x. 16 / 26

Idea of the proof • From the concavity property one can prove that, for some x ( t ) ∈ R N , n ε − ⇀ n , supp n = { x ( t ) } . • The case with constant environment ( R ( x , I ) ): There exists a unique constant I ( t ) such that R ( x ( t ) , I ( t )) = 0 . One can then prove that, as ε → 0, I ( t ) I ε ( t ) − → I ( t ) , n ε − ⇀ η ( x ( t )) δ ( x − x ( t )) . 17 / 26

• Periodic environment ( R ( x , s , I ) ): more work to determine the ( weak ) limit of I ε . Lemma (Cell problem) Let x = x ( t ) for some t ≥ 0 . There exists a unique 1 -periodic positive solution I ( x , s ) : [ 0 , 1 ] → ( 0 , I M ) to  d � � ds I ( x , s ) = I ( x , s ) R x , s , I ( x , s ) ,  I ( x , 0 ) = I ( x , 1 ) .  Then � 1 I ε ( t ) − ⇀ I ( x ( t ) , s ) ds , 0 � 1 R ( x , t ε, I ε ( t )) − ⇀ R ( x , x ( t )) := R ( x , s , I ( x ( t ) , s )) ds . 0 18 / 26

Weak convergence of I ε 8 7 6 5 I 4 3 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t Dynamics of the total population I ε ( t ) with R ( x , s , I ) = ( 2 + sin ( 2 π s )) 2 − x 2 I + . 5 − . 5 , η ( x ) = 1 , ε = 0 . 01 . The I ε ’s oscillate with period of order ε around a monotone curve I . 19 / 26