Matrix Models for Evolutionary Population Dynamics: Studies of the Effects of Climate Change on Seabirds

Matrix Models for Evolutionary Population Dynamics: Studies of the Effects of Climate Change on Seabirds J. M. Cushing Department of Mathematics & Interdisciplinary Program in Applied Mathematics University of Arizona In collaboration with co-pi s Shandelle Henson & James Hayward Departments of Mathematics & Biology, Andrews University and my students Kehinde Salau Allianz Post-doctoral Fellow University of Arizona Amy Veprauskas Emily Meissen Interdisciplinary Program in Applied Mathematics University of Arizona With the support from Mathematical Biology Program Division of Mathematical Sciences Population & Community Ecology Program Division of Environmental Biology

Established 1988 Vancouver Island Canada Olympic Peninsula Washington State Seattle

Glaucous-winged Gulls Sentinels of climate change Hosts 70,000 nesting seabirds

Mean sea surface temperature (MST) increases during El Niño years. Increased MST effects on gull s marine food resource availability. Numerous behavioral changes have been observed in El Niño years. These include: reproductive timing, cannibalism, length of breeding season Long term climate trend: MST is increasing. What effects do these changes have on o Long-term population dynamics? o Are the populations endangered? o Are the changes adaptive?

One component of our research is to use (low dimensional) proof-of-concept models to investigate various hypotheses concerning these questions In this talk, I will describe a model designed to investigate the population & evolutionary dynamic consequences of cannibalism & reproductive timing ( in the presence of environmental deterioration )

Modeling Methodology o Matrix models for dynamics of structured populations o Evolutionary x game theoretic (EGT) versions of matrix models 1 x : non-negative class distribution vector x m (stage, age, size, etc.) x = distribution vector at next census time x = P x x The m m projection matrix P x is non-negative & irreducible. Its entries of describe : fertility rates survival & stage transition probabilities cannibalism rates, etc.

Modeling Methodology o Matrix models for dynamics of structured populations o Evolutionary x game theoretic (EGT) versions of matrix models 1 x : non-negative class distribution vector x m (stage, age, size, etc.) x = distribution vector at next census time x = P x x 0 < r x = spectral radius of P(x) = dominant eigenvalue ( Perron-Frobenius theory )

Modeling Methodology o Matrix models for dynamics of structured populations o Evolutionary game theoretic (EGT) versions of matrix models v = a phenotypic trait of an individual (subject to Darwinian evolution) u = mean phenotypic trait of the population Assume entries of P = [p ij (x,v,u)] depend on x, v and u.

x P( x, v, u) v u x 2 u u F x v u v (,, ) v u Population Dynamics Trait Dynamics T. Vincent & J. Brown Cambridge U Press 2005

x P( x, v, u) v u x 2 u u F x v u v (,, ) v u Population Dynamics Trait Dynamics Fitness usually F(x,v,u) = ln r(x,v,u)

x P( x, v, u) v u x 2 u u r x v u v ln (,, ) v u Population Dynamics Trait Dynamics Trait variance = speed of evolution

x P( x, v, u) v u x 2 u u r x v u v ln (,, ) v u Population Dynamics Trait Dynamics Known by many names: Fisher s equation, Lande s equation, the breeder s equation, or the canonical equation of evolution

x P( x, v, u) v u x 2 u u r x v u v ln (,, ) v u Population Dynamics Trait Dynamics Extinction Equilibria (x,u) = (0, u*) where u* is a critical trait 0 vr : v r(0, v, u*) vu* 0 A basic question: Is an extinction equilibrium stable or unstable?

x P( x, v, u) v u x 2 u u r x v u v ln (,, ) v u Population Dynamics Trait Dynamics The Linearization Principle If r 0 then (0, u*) is unstable. 0 vv

x P( x, v, u) v u x 2 u u r x v u v ln (,, ) v u Population Dynamics Trait Dynamics 0 If r The Linearization Principle If r 0 then (0, u*) is unstable. 0 vv vv we get that 0 then defining r : r(0, u*) 0 r r 0 0 1 (0, u*) 1 (0, u*) is (locally asymptotically) stable is unstable

x P( x, v, u) v u x 2 u u r x v u v ln (,, ) v u Population Dynamics Trait Dynamics Assume vv 0 r < 0. What kind of bifurcation occurs at r 0 = 1?

x P( x, v, u) v u x 2 u u r x v u v ln (,, ) v u Population Dynamics Trait Dynamics Assume vv 0 r < 0. What kind of bifurcation occurs at r 0 = 1? If P(0, u*, u*) is primitive, the answer is known: There exists a bifurcating continuum of positive equilibria which are stable if the bifurcation is forward (r 0 > 1 ) unstable if the bifurcation is backward (r 0 < 1). JMC, Martins, Pinto, Veprauskas, J. Math. Biology (2017) Meissen, Salau, JMC, J. Diff. Eqns Appl. (2016) JMC, Nonlinear. Studies (2010)

x P( x, v, u) v u x 2 u u r x v u v ln (,, ) v u Population Dynamics Trait Dynamics If P(0, u*, u*) is primitive, the answer is known: There exists a bifurcating continuum of positive equilibria, which are stable if the bifurcation is forward (r 0 > 1 ) unstable if the bifurcation is backward (r 0 < 1). stable Assume 0 vv r < 0. What kind of bifurcation occurs at r 0 = 1? JMC, Martins, Pinto, Veprauskas, J. Math. Biology (2017) Meissen, Salau, JMC, J. Diff. Eqns Appl. (2016) JMC, Nonlinear. Studies (2010) ( xu, ) ( xu, ) unstable (0, u*) stable 1 unstable r 0 (0, u*) stable 1 unstable r 0

Juveniles A Cannibalism-Reproductive Synchrony Model s1 2 Reproductively active adults s Reproductively inactive adults x 1 b x 2 s3 g x 3 0 b 0 P( x) s1 0 s3g 0 s2 s3(1 g x x x x 1 2 3 s ( g) 3 1

Juveniles A Cannibalism-Reproductive Synchrony Model s1 2 Reproductively active adults s Reproductively inactive adults x 1 b x 2 s3 g x 3 0 b 0 P( x) s1 0 s3g( x2 ) 0 s2 s3(1 g( x2 )) x x x x 1 2 3 s ( g) 3 1 Larger g( x ) decreasing function with g(0) 1 2 g(0) means higher propensity to synchronize Example : g( x2) exp( cx2)

A Cannibalism-Reproductive Synchrony Model 0 b p12( x) 0 P( x) s1 p21( x) 0 s3 p23( x) 0 s2 p32( x) s3 p33( x) p (0) 1, p ( x) 0 when x 0 ij 33 2

A Cannibalism-Reproductive Synchrony Model 0 b p12( x) 0 P( x) s1 p21( x) 0 s3 p23( x) 0 s2 p32( x) s3 p33( x) p (0) 1, p ( x) 0 when x 0 ij 33 2 0 b 0 P(0) s1 0 s3 0 s2 0 IMPRIMITIVE! eigenvalues bs s 2, 0 1 3 r bs s 2 0 1 3 is not strictly dominant

x P( x, v, u) v u x 2 u u r x v u v ln (,, ) v u Population Dynamics Trait Dynamics Assume vv 0 r < 0. What kind of bifurcation occurs at r 0 = 1?

x P( x, v, u) v u x 2 u u r x v u v ln (,, ) v u Population Dynamics Trait Dynamics Assume 0 vv r < 0. What kind of bifurcation occurs at r 0 = 1? If P(0, u*, u*) is imprimitive, only some answers are known for specialized matrices P.

x P( x, v, u) v u x 2 u u r x v u v ln (,, ) v u Population Dynamics Trait Dynamics Assume vv 0 r < 0. What kind of bifurcation occurs at r 0 = 1? If P(0, u*, u*) is imprimitive, only some answers are known for specialized matrices P. For example, there is a large literature on semelparous Leslie matrices. A REVIEW: JMC & Henson, Stable bifurcations in nonlinear semelparous Leslie models Journal of Biological Dynamics 6 (2012), 80-102 Another example: synchronous matrix models A. Veprauskas On the dynamic dichotomy between positive equilibria and synchronous 2-cycles in matrix population models, PhD dissertation, U of Arizona (2016)

A Cannibalism-Reproductive Synchrony Model 0 b( v) p12( x) 0 P( x, v) s1( v) p21( x, v) 0 s3( v) p23( x, v) 0 s2( v) p32( x, v) s3( v) p33( x, v) p (0, v) 1, p ( x, v) 0 when x 0 ij 33 2 THEOREM Veprauskas & JC, J. Biological Dynamics 2016 Veprauskas, PhD dissertation, U of Arizona (2016) If 0 vv r < 0, then (0, u*) loses stability as r 0 increases through 1. Two unbounded continua bifurcate from x = 0 at r 0 = 1 : o positive equilibria o synchronous 2-cycles.

For this matrix model 1. A portion of the boundary of the positive cone is invariant. x 3 (x 2, x 3 )-plane x 2 -axis x 1 2. Orbits on this portion of the boundary have the form: 0 0 0 0 0 0 are called synchronous orbits. Note reproductive synchrony!

A Cannibalism-Reproductive Synchrony Model 0 b( v) p12( x) 0 P( x, v) s1( v) p21( x, v) 0 s3( v) p23( x, v) 0 s2( v) p32( x, v) s3( v) p33( x, v) p (0, v) 1, p ( x, v) 0 when x 0 ij 33 2 p 0 ij k pij : x Diagnostic quantities cb : (1 s s ) p p s s p p s s s p 0 0 0 0 2 0 2 3 12 13 21 2 2 3 23 2 32 13 1 2 3 2 33 k x0, vu* r 1 T 0 s 0, 1 s s 0 s s 0 low density sensitivities 2 1 13 2 3 1 2 Weighted sum of BETWEEN-category low density sensitivities (reproductive vs non-reproductive classes) T

A Cannibalism-Reproductive Synchrony Model 0 b( v) p12( x) 0 P( x, v) s1( v) p21( x, v) 0 s3( v) p23( x, v) 0 s2( v) p32( x, v) s3( v) p33( x, v) p (0, v) 1, p ( x, v) 0 when x 0 ij 33 2 p 0 ij k pij : x k x0, vu* r 1 0 Diagnostic quantities low density sensitivities cb : (1 s s ) p p s s p p s s s p 0 0 0 0 2 0 2 3 12 13 21 2 2 3 23 2 32 13 1 2 3 2 33 0 0 0 0 cw : (1 s2s3 ) p12 2 p21 13 s2s 3 p23 13 p32 2 Weighted sum of WITHIN-category low density sensitivities

A Cannibalism-Reproductive Synchrony Model 0 b( v) p12( x) 0 P( x, v) s1( v) p21( x, v) 0 s3( v) p23( x, v) 0 s2( v) p32( x, v) s3( v) p33( x, v) p (0, v) 1, p ( x, v) 0 when x 0 ij 33 2 p 0 ij k pij : x k x0, vu* r 1 Diagnostic quantities a : c c w a : c c 0 w low density sensitivities cb : (1 s s ) p p s s p p s s s p 0 0 0 0 2 0 2 3 12 13 21 2 2 3 23 2 32 13 1 2 3 2 33 0 0 0 0 cw : (1 s2s3 ) p12 2 p21 13 s2s 3 p23 13 p32 2 b b

Positive equilibria a + > 0 a + < 0 Direction of bifurcation Stability Properties A Cannibalism-Reproductive Synchrony Model 0 b( v) p12( x) 0 P( x, v) s1( v) p21( x, v) 0 s3( v) p23( x, v) 0 s2( v) p32( x, v) s3( v) p33( x, v) p (0, v) 1, p ( x, v) 0 when x 0 ij THEOREM Assume a + 0 and c w 0. Backward 33 2 Forward Unstable Stable if a - < 0 Unstable if a - > 0

THEOREM Assume a + 0 and c w 0. Positive equilibria a + > 0 a + < 0 Direction of bifurcation Stability Properties Backward Forward Unstable Stable if a - < 0 Unstable if a - > 0 Synchronous 2-cycles c w > 0 c w < 0 Direction of bifurcation Stability Properties A Cannibalism-Reproductive Synchrony Model 0 b( v) p12( x) 0 P( x, v) s1( v) p21( x, v) 0 s3( v) p23( x, v) 0 s2( v) p32( x, v) s3( v) p33( x, v) p (0, v) 1, p ( x, v) 0 when x 0 ij Backward 33 2 Forward Unstable Stable if a - > 0 Unstable if a - < 0

A Cannibalism-Reproductive Synchrony Model 0 b( v) p12( x) 0 P( x, v) s1( v) p21( x, v) 0 s3( v) p23( x, v) 0 s2( v) p32( x, v) s3( v) p33( x, v) p (0, v) 1, p ( x, v) 0 when x 0 ij 33 2 THEOREM Assume a + 0 and c w 0. Positive equilibria a + > 0 a + < 0 Direction of bifurcation Stability Properties Backward Forward Unstable Stable if a - < 0 Unstable if a - > 0 Synchronous 2-cycles c w > 0 c w < 0 Direction of bifurcation Stability Properties Backward Forward Unstable Stable if a - > 0 Unstable if a - < 0 Punch lines A Dynamic Dichotomy Backward bifurcations are unstable and caused by positive feedbacks at low density (component Allee effects)

The Usual Properties of Backward Bifurcations JMC, Journal of Biological Dynamics (2014) JMC, Applied Analysis with Applications in Biological & Physical Sciences (2016) stable Negative density effects dominate at high density unstable (0, u*) stable unstable 1 r 0

The Usual Properties of Backward Bifurcations JMC, Journal of Biological Dynamics (2014) JMC, Applied Analysis with Applications in Biological & Physical Sciences (2016) SIGNIFICANCE stable Possibility of survival when r 0 < 1. Initial condition dependent: Endangered by the basin of attraction of the extinction equilibrium Creates a tipping point, at which a sudden collapse to extinction occurs (0,u*) stable unstable unstable 1 An interval of Strong Allee effects r 0

An Example Cannibalism-Reproductive Synchrony Model 0 b p12( x) 0 P( x) s1 p21( x) 0 s3 p23( x) 0 s2 p32( x) s3 p33( x) p (0) 1, p ( x) 0 when x 0 ij 33 2 A function of a phenotypic trait v 1 Example: b( v) b0 2 c v Food resource availability Foraging rate is maximized at trait v = 0

An Example Cannibalism-Reproductive Synchrony Model 0 b( v) p12( x) 0 P( x, v) s1 p21( x) 0 s3 p23( x) 0 s2 p32( x) s3 p33( x) p (0, v) 1, p ( x) 0 when x 0 ij 33 2 Decreasing in x i (Density regulation of fertility) Example: 1 1 cx 2

An Example Cannibalism-Reproductive Synchrony Model 0 b( v) p12( x) 0 P( x, v) s1 p21( x) 0 s3 p23( x) 0 s2 p32( x) s3 p33( x) p (0, v) 1, p ( x) 0 when x 0 ij 33 2 Decreasing in x 2 and x 3 ( Cannibalism ) Increasing in x 1 ( Victim saturation effect) Example: wx i i 1 1 1 w x 1 cx i i 1 Holling Type II victim uptake rate Victim saturation of cannibals

An Example Cannibalism-Reproductive Synchrony Model 0 b( v) p12( x) 0 P( x, v) s1 p21( x, v) 0 s3 p23( x) 0 s2 p32( x) s3 p33( x) p (0, v) 1, p ( x) 0 when x 0 ij 33 2 Cannibalism intensity a function of a phenotypic trait v, negatively correlated with b(v). Example: wi( v) xi 1 1 1 w ( v ) x 1 cx i i 1 Cannibalism intensity is minimized at trait v = 0 1 v wi( v) wi c 2 v 2

An Example Cannibalism-Reproductive Synchrony Model 0 b( v) p12( x) 0 P( x, v) s1 p21( x, v) 0 s3 p23( x, v) 0 s2 p32( x, v) s3 p33( x, v) Increasing functions of the number of juveniles cannibalized wx1 1 i Example: where ( z) is increasing 1wixi 1w1x1

A Cannibalism-Reproductive Synchrony Model x1 0 b( u) p12( x) 0 x1 x s p ( x, u) 0 s p ( x, u) x 2 1 21 3 23 2 x 3 0 s2 p32( x, u) s3 p33( x, u) x 3 2 u u r x v v ln (, ) v u r(0, v) b( v) s s s 1 2 3 Critical traits: 0 v r 0 b ( u *) 0 u * 0 s 0 1 vvr 2 b(0) 0 b r s s s c 0 0 1 2 3

= 0.5 r 0 = 0.96 = 0.7 r 0 = 1.09 = 0.8 r 0 = 1.15 Severely degraded environment Degraded environment Healthy environment Food resource availability decreases

= 0.5 r 0 = 0.96 = 0.7 r 0 = 1.09 = 0.8 r 0 = 1.15 Evolution selects against (eliminates) cannibalism

= 0.5 r 0 = 0.96 = 0.7 r 0 = 1.09 = 0.8 r 0 = 1.15 Decreased food resource increases cannibalism intensity Victim saturation promotes reproductive synchrony Evolution selects for a low level of cannibalism Low level of Cannibalism Cannibalism eliminated

= 0.5 r 0 = 0.96 = 0.7 r 0 = 1.09 = 0.8 r 0 = 1.15 Higher level of Cannibalism Low level of Cannibalism Cannibalism eliminated

= 0.5 r 0 = 0.96 = 0.7 r 0 = 1.09 = 0.8 r 0 = 1.15 Diagnostics: a + = 0.07 > 0, c w = 0.06 > 0 imply a backward bifurcation

= 0.5 r 0 = 0.96 What happens if this population cannot adapt? Change 2 = 0.01 to 2 = 0 and the population goes extinct.

= 0.5 r 0 = 0.96 What happens if this population cannot adapt? Change 2 = 0.01 to 2 = 0 and the population goes extinct:

Take home messages for this model population Evolution can select for cannibalism in a deteriorated environment (decreased in environmental food resource). Cannibalistic population can survive in a deteriorated environment in which a non-cannibal cannot. Victim saturation as a protection against cannibalization can lead to reproductive synchrony & non-equilibrium dynamics.

Ongoing & future projects include o o Non-constant environments Periodic, stochastic Across season model to account for different maturation & reproductive synchrony time scales o Add predation Bald eagle predation can be significant in gull colonies o Evolution of multiple traits