Monotone Dynamical Systems: Reflections on New Advances ...halsmith/AIMS.pdf · Applications: consensus in multi-agent networks, ecological systems, in-vivo disease modeling, chemical

Monotone Dynamical Systems: Reflections on NewAdvances & Applications

Hal Smith

A R I Z O N A S T A T E U N I V E R S I T Y

H.L. Smith Monotone Systems AIMS, Orlando July 5, 2016 1 / 50

SS15: Session on Monotone Dynamical Systems &Applications, J. Mierczynski & S.

Morris W Hirsch, Monotone semidynamical systems with dense periodic pointsMats Gyllenberg, Group actions on monotone skew-product semiflowsYuan Lou, Dispersal in advective environmentsSze-Bi Hsu, Competition for two essential resources with internal storage and periodic inputJianhong Wu, Monotone semiflows with respect to high-rank cones on a Banach spaceKing-Yeung Lam, Persistence Results of a PDE Population Model of Phytoplankton with Ratio DependenceStephen Baigent, Carrying simplicies of continuous and discrete-time systemsFulvio Forni, Open differential positive systems: attractors and interconnectionYi Wang, On heteroclinic cycles of competitive maps via carrying simplices, PS13, 13:30-14:00Xingfu Zou, Asymptotic behaviour, spreading speeds and traveling waves of some dynamical systems, PS13, 14:00-14:30Jian Fang, Bistable Traveling Waves for Monotone Semiflows, PS13, 14:30-15:00Patrick De Leenheer, Persistence of aquatic insect populations subject to flooding, PS13, 15:00-15:30Jifa Jiang, The Decomposition Formula for Stochastic Lotka-Volterra Systems with Identical Intrinsic Growth Rate and itsApplications to Stationary Motions, PS14, 16:00-16:30Wenxian Shen, Criteria for the Existence of Principal Eigenvalue of Time Periodic Cooperative Linear Systems with NonlocalDispersal, PS14, 16:30-17:00

Room: Celebration 6


Outline

What is a monotone dynamical system?

A gentle introduction to continuous-time, autonomous, monotonesystems.

Examples of monotone systems, mainly from biological sciences.What’s new in the theory & applications?

1 An equation-free theory of two-species competition.2 Monotone control theory.

Caveat: very little name dropping, most general results not stated, only simple-to-describe applications mentioned


Monotone Dynamical Systems begin with M.W. HirschInfluential Papers from 1980s

Systems of differential equations which are competitive or cooperative 1: limit sets.

Systems of differential equations which are competitive or cooperative II: convergence almost everywhere.

Systems of differential equations which are competitive or cooperative III, Competing species.

Systems of differential equations that are competitive or cooperative. IV: Structural stability in three dimensional systems.

Systems of differential equations that are competitive or cooperative. V: Convergence in 3-dimensional systems.

Stability and Convergence in Strongly Monotone dynamical systems.


Monotone Dynamics: an active research area

Monotone Mappings, Discrete-time systems.

Non-autonomous monotone systems: skew-product dynamicalsystems, random dynamical systems.

Traveling waves

MathSciNet journal articles since 2000: 106 papers list key word “monotone system”,

225 list “cooperative systems”.

Applications: consensus in multi-agent networks, ecological systems, in-vivo disease modeling, chemical

reaction dynamics, epidemiology, systems science/gene-networks, population genetics, neural networks, physiological

modeling.


Autonomous Monotone Dynamical Systems

State Space: X , a subset of ordered Banach space Y with cone Y+:

x ≤ y ⇔ y−x ∈ Y+, x < y ⇔ y−x ∈ Y+\0, x ≪ y ⇔ y−x ∈ IntY+.

Y+ is closed, convex, [0,∞) · Y+ ⊂ Y+. Example: Y = C(A,R), Y+ = y ∈ Y : y(a) ≥ 0, a ∈ A

Time Set: T = 0,1,2, · · · or T = [0,∞).

Semiflow: Φ : X × T → X where Φt(x) = Φ(x , t) satisfies

Φ0 = I, Φt Φτ = Φt+τ , t , τ ∈ T .

Φ is continuous.

Order Preserving Semiflow: x ≤ y ⇒ Φt(x) ≤ Φt(y), t ∈ T .






Time Set: T = 0,1,2, · · · or T = [0,∞).


Φ0 = I, Φt Φτ = Φt+τ , t , τ ∈ T .

Φ is continuous.







Time Set: T = 0,1,2, · · · or T = [0,∞).


Φ0 = I, Φt Φτ = Φt+τ , t , τ ∈ T .

Φ is continuous.







Time Set: T = 0,1,2, · · · or T = [0,∞).


Φ0 = I, Φt Φτ = Φt+τ , t , τ ∈ T .

Φ is continuous.



Examples of Order-Preserving Dynamics

The simplest monotone dynamics

Bigger initial data imply bigger future states.

x ′ =dxdt

= f (x), f : R → R

0 0.5 1 1.5 2 2.5

time t

0

0.2

0.4

0.6

0.8

1

1.2

x(t)

One-dimensional dynamics is order preserving



Cooperative System of ODEs x ′ = f (x)

Component-wise partial order:1 x ≤ y ⇔ ∀i , xi ≤ yi .2 x < y ⇔ x ≤ y & x 6= y .3 x ≪ y ⇔ ∀i , xi < yi .

x1

x2

x•

y•

cone=Rn+

Quasi-monotone condition: xj → fi(x) is nondecreasing for i 6= j .It holds if f ∈ C1(D) with convex D and ∂fi

∂xj(x) ≥ 0, i 6= j ; we say the

system is cooperative.

A cooperative system is order-preserving:x0 <r x0 ⇒ x(t , x0) <r x(t , x0), t > 0, <r=≤, <,≪ .



Example: single species migrating between patches

xi = density of species in patch i

x ′

i = rixi(1 − xi/Ki) +∑

j

djixj −∑

j

dijxi , 1 ≤ i ≤ n

where dij = rate of migration from patch i to patch j .



Additive Cooperative Neural Networks

xi = i-th neuron activity level

x ′

i = −Aixi +

n∑

j=1

tanh(xj)Wij + Ii

Wij = weight of connection j to i

If matrix Wij ≥ 0, i 6= j , then the system is cooperative.



Ross-Macdonald multi-patch malaria model

xi (yi ) represents the proportion of infected human (infected mosquito)residents of patch i .

x ′

i =

n∑

j=1

ρ−1i pijρjR

0j α

−1j rjyj

(1 − xi)− rixi

y ′

i =

n∑

j=1

qijαjµjxj

(1 − yi)− µiyi , 1 ≤ i ≤ n,

where pij (qij ) is the fraction of time a human (mosquito) resident ofpatch i spends in patch j .

X = (x, y) ∈ R2n : 0 ≤ xi , yi ≤ 1. P. de Leenheer et al.



Orthant Cone Rk+ × (−Rn−k

+ ), 1 ≤ k ≤ n

x = (x1, x2) ≤C (x1, x2) = x ⇔ x1 ≤ x1 ∧ x2 ≥ x2.

Seek conditions for order preservation: x0 ≤C x0 ⇒ x(t , x0) ≤C x(t , x0), t ≥ 0.

x1 ∈ Rk

x2 ∈ Rn−k

x•

x•

Cone Rk+ × (−Rn−k

+ )



Generalized Cooperative System: competition between two teams

Suppose x ′ = f (x) can be decomposed x = (x1, x2) ∈ Rk × Rn−k

x ′

1 = f1(x1, x2)

x ′

2 = f2(x1, x2)

diagonal blocks ∂fi∂xi

(x) have nonnegative off-diagonal entries.

off-diagonal blocks ∂fi∂xj

(x) ≤ 0 have nonpositive entries.

Jacobian of f =

∗ + − −+ ∗ − −− − ∗ +− − + ∗

Components cluster into two subgroups. positive within-group interactions,negative between-group interactions. quasi-monotone case when k = n



Example: 2-gene repressilator is cooperative

xi = [protein] product of gene iyi = [mRNA] of gene i.xi represses transcription of yj , i 6= j:

x ′

1 = β1(y1 − x1)

y ′

1 = α1f1(x2)− y1

x ′

2 = β2(y2 − x2)

y ′

2 = α2f2(x1)− y2

fi > 0 & f ′i < 0.

G1 G2

Jacobian =

∗ + 0 00 ∗ − 00 0 ∗ +− 0 0 ∗

Gardner et al, “Construction of a genetic toggle switch in E. coli", Nature(403),2000.



Repressilator with translation/transcription delays

x ′

i (t) = βi [yi(t − µi)− xi(t)]

y ′

i (t) = αi fi(xi−1(t − τi−1))− yi(t), i = 1,2

The state space is

X = C([−τ1,0],R+)×C([−µ1,0],R+)×C([−τ2,0],R+)×C([−µ2,0],R+).

The competitive ordering on X is defined by

(x1, y1, x2, y2) ≤C (x1, y1, x2, y2) ⇔ x1 ≤ x1, y1 ≤ y1 ∧ x2 ≥ x2, y2 ≥ y2

It is preserved by the dynamics of the delayed repressilator.



Neanderthals versus modern humans

Neanderthal density N1 and their cultural level z1;N2, z2 denote corresponding values for modern humans:

N ′

i = riNi

(

1 −Ni + bijNj

Mi(zi )

)

z′

i = −γizi + δiNi , i 6= j .

carrying capacity Mi(zi ) increases with cultural level zi .

higher cultural level hominid exerts greater competitive effect on lowercultural level rival:

bij = b0(

1 + ǫ(zj − zi))

, 0 < ǫ ≪ 1,

Gilpin,Feldman,Aoki, An ecocultural model predicts Neanderthal extinction through competition with modern humans, PNAS,2016



How we won: purple N20/N10 = 0.7; blue N20/N10 = 0.9



A test for generalized cooperativity of x ′ = f (x)

Unambiguous influence: ∀i 6= j , ∂fi∂xj

(x) does not change sign.

Feedback Symmetry: ∂fi∂xj

(x) ∂fj∂xi

(y) ≥ 0, i 6= j . golden rule

Construct signed, influence graph:un-directed edge joins i to j 6= i if ∃x ∈ X , ∂fi

∂xj(x) 6= 0.

append + sign to edge if derivative is positive, − sign if negative.

balanced graph (‡): every loop (cycle) has even number of “−"signs.

‡ This is Harary’s Theorem: “a balanced network is clusterable". See “Networks: An Intro.", M. NewmanAn algorithm is given for clustering, i.e, permuting indices into subsets I = 1, 2, · · · , k and Ic .



Systems Biology: source of cooperative systems

Network structure may be all that is known.



Reaction-Diffusion with cooperative reaction term

∂ui

∂t= di∆ui + f (x ,u), x ∈ Ω, t > 0,

∂νui = 0, x ∈ ∂Ω, t > 0,

u(x , t) = u0(x), x ∈ Ω,

where di > 0, ν is outward normal, Ω ⊂ Rn, u = (u1, · · · ,um) ∈ Rm+.

fi(x ,u) ≥ 0 when u ≥ 0 and ui = 0.u → f (x ,u) satisfies cooperativity condition with orthant O ⊂ Rm.

State space: X = C(Ω,Rm+) in Y = C(Ω,Rm).

Cone: Y+ = C(Ω,O).

Maximum Principle implies that semiflow Φt(u0) = u(·, t) isorder-preserving w.r.t. Y+.Dirichlet or Robin boundary conditions may also be considered; general elliptic diff. operator.



Example: Model of “river drift paradox*”

How does a “drunk” animal population persist in a rapidly flowing river?

v = moving phase, w = reproducing phase of population

vt = vxx − νvx − pv + qw , x ∈ R,or, x ∈ [0,L]

wt = w(1 − w) + pv − qw

where ν = advection velocity. Bdry. Cond. : vx (0) = 0, v(L) = 0.

1 the system preserves usual pointwise order.

2 q < 1 ⇒ persistence.3 if q > 1, ν < ν∗(p, q) is necessary for persistence.

4 L > L∗(p, q, ν) sufficient for persistence.

*Pachepsky, Lutscher, Nisbet, Lewis, TPB 2005


General Theory of Autonomous Order-Preserving Dynamics

How does monotonicity impact dynamics?

1 Comparison principles apply, e.g.Φt(x) ր p ⇒ Φt(y) → p, x < y < p.

2 t → Φt(x) cannot have both a rising interval and a falling interval.[a, b] is rising if Φa(x) < Φb(x).

3 A nontrivial periodic orbit cannot be attracting if T = [0,∞).4 Except in very low dimensions, monotonicity alone is too weak for

optimal results.



Strongly monotone semiflow

Φ is strongly monotone (SM) if:

x < y ⇒ Φt(x) ≪ Φt(y), i .e. Φt(y) − Φt(x) ∈ IntY+, t > 0.

x1

x2

x•

y• Φt(x)•

Φt(y)•

In applications to cooperative ODEs & PDEs, strong monotonicityholds if the Jacobian of f is an irreducible matrix.



Convergent & Quasiconvergent points

Equilibria:E = x ∈ X : Φt(x) = x , t ≥ 0.

Omega Limit Set:

ω(x) = y ∈ X : ∃tn → ∞,Φtn(x) → y

Quasi-convergent points:

Q = x ∈ X : ω(x) ⊂ E.

Convergent points:

C = x ∈ Q : ω(x) is a singleton.



Generic Quasi-Convergence

Theorem : Let Φ be SM with pre-compact orbits.(i) If X is a convex subset of an ordered Banach space, then Q is

residual in X .(ii) If every point of X can be approximated either from above or from

below and if Φ satisfies compactness condition (C), then Int Q isdense in X and X = Q if E is a singleton.

(iii) If X ⊂ C(A,Rn) where A is a compact Hausdorff space and theorder relation is the restriction to X of the usual pointwise orderingwhere Rn is ordered by an orthant cone, then Int Q is dense in X .

(iv) If X ⊂ Y where Y is a separable Banach space, then Int Q isprevalent* in X .

*Borel set S is shy if ∀y ∈ Y , µ(y + S) = 0 for some non-zero compactly supported Borel measure µ. S is prevalent if its

complement is shy. Hunt, Sauer, Yorke 1992. Prevalence: a translation invariant notion of “almost everywhere”.

Hirsch (i); S.& Thieme (ii); Hirsch & S. (iii); Enciso, Hirsch, S. (iv)

Q → C with additional hypotheses



3 Key Tools in Proof

Theorem [Hirsch]: Let Φ be SM with precompact orbits. Then(a) Convergence Criterion: If Φτ (x) > x or Φτ (x) < x for some τ > 0

then Φt(x) → p ∈ E .(b) Non-ordering of Limit Sets: No 2 points of an omega limit set are

related by <.(c) Limit Set Dichotomy: If x < y then either ω(x) ≪ ω(y), or

ω(x) = ω(y) ⊂ E .

y••x cone = R2

+

ω(y)

ω(x)



Complex dynamics?

[Smale, 1976] Any smooth dynamical system on Σn−1 = xi ≥ 0,∑

xi = 1can be extended to a competitive, dissipative Kolmogorov vector field on Rn

+:

x ′

i = xi fi (x),∂fi∂xj

≤ 0, i 6= j

making Σn−1 an attractor bounding basin of repulsion of zero and infinity.



Time reversal t → −t : competitive system becomes cooperative

Any smooth dynamical system on Σn−1 = xi ≥ 0,∑

xi = 1 can beextended to a cooperative, dissipative Kolmogorov vector field on Rn

+:

x ′

i = xi fi(x),∂fi∂xj

≥ 0, i 6= j

making Σn−1 an repellor bounding basin of attraction of zero andinfinity.

Complex dynamics are restricted to basin boundaries of attracting equilibria.



Complex dynamics are unstable

Thm. [Hirsch,Takac] Let Φ be SM, Φt compact, and compact sets havebounded orbits. If ω(x) * E , ∃p,q ∈ E such that p ≪ ω(x) ≪ q and

1 ω(y) = p if p < y < w , some w ∈ ω(x).2 ω(y) = q if w < y < q, some w ∈ ω(x).

ω(x) ⊂ H, an unordered, positively invariant, co-dimension one,Lipschitz manifold.

Smale’s construction captures the generic case.


A theory of competition between 2 agents

Abstract Theory of competition between 2 agents

models of two-agent competitionappear in a wide range ofmathematical forms.

the dynamics of these modelsshare common features.

can one develop an equation-freetheory of competition?

hypotheses based solely onproperties of single-agentdynamics.

P. Hess & A. Lazer, 1991Ellermeyer, Hsu, Waltman, 1994Hsu, S., Waltman, 1996S. & Thieme , 2001Liang & Jiang, 2002Lam & Munther, 2016



State Space and Order Relation for Competition

Let Xi , i = 1,2, be ordered Banach spaces with positive cones X+i and

denote by IntX+i , the interior of X+

i .X+

i is the state space for competitor i .

Let X = X1 × X2 with cone X+ = X+1 × X+

2 .X+ represents the state space for two species competition.x = (x1, x2) ∈ X+

The dynamics should preserve the competition order relation

x ≤C x ⇐⇒ x1 ≤ x1 and x2 ≤ x2

generated by the “south-east” cone C = X+1 × (−X+

2 ).



Hypotheses for Competition

(H1) Φ is SM with the order <C , and Φt is compact mapping for t > 0.

(H2) E0 = (0,0) is a repelling equilibrium.

(H3) Unique single-species equilibria Ei , i = 1,2 attract all nonzeropoints on their respective axes.

E2 •

E1

••

E0

predict dynamics in IntX+.

X+2

X+1



Attracting invariant Interval I = x : E2 ≤C x ≤C E1.

E2 •

E1

••

E0

(x1, x2) = x•(0, x2)•

(x1,0)

•

Iω(x)

(0, x2) <C x <C (x1,0) ⇒ Φt(0, x2) <C Φt(x) <C Φt(x1,0).



Competitive Exclusion or Coexistence Equilibrium

Theorem:[Hsu,S.,Waltman] If there is no coexistence equilibrium belonging toI ∩ IntX+ then exactly one of the following holds:

(a) Φt(x) → E1 for all x = (x1, x2) ∈ I with xi > 0, i = 1, 2.

(b) Φt(x) → E2 for all x = (x1, x2) ∈ I with xi > 0, i = 1, 2.

If (a) or (b) hold and x /∈ I, then either Φt (x) → E1 or Φt(x) → E2.

E2 •

E1

••

E0

heteroclinic orbit E2 •

E1

••

E0

heteroclinic orbit



Sufficient conditions for coexistence

Corollary: There is a coexistence equilibrium in I ∩ IntX+ if:

(i) Both E1 and E2 are stable relative to I, or

(ii) Both E1 and E2 are unstable relative to I, or

(iii) There is a point x ∈ X+ and a point z ∈ ω(x) such that z ∈ IntX+.



Persistence/Coexistence

Theorem:[Hess & Lazer] If E1 and E2 are isolated and unstable relativeto I, then there exist equilibria E∗,E∗∗ ∈ I ∩ IntX+, possibly identical,such that

E2 <<C E∗∗ ≤C E∗ <<C E1

andE∗∗ ≤C ω(x) ≤C E∗, ∀x = (x1, x2) ∈ I, xi 6= 0.

E2 •

E1

••

E0

•

•

E∗∗

E∗

ω(x)



[S.& Thieme]: Bi-Stability and Basins of Attraction

Theorem: Assume a unique equilibrium E∗ ∈ I ∩ IntX+ and it’s a saddle point:Φt0

is C1 on a neighborhood of E∗ and the spectral radius of DxΦt0(E∗) is strictly greater than one for some t0 > 0.

Let Bi = x ∈ X+ : ω(x) = Ei, B∗ = x ∈ X+ : ω(x) = E∗.Then:

(a) x ∈ X+ : x <C E∗ ⊂ B2 and x ∈ X+ : E∗ <C x ⊂ B1.

(b) S = X+ \ (B1 ∪ B2) is <C-unordered, positively invariant set consisting ofE0, B∗, and a possibly empty set of non-quasi-convergent points.

(c) B1 ∪ B2 is open & dense in X+; S is Lipschitz co-dim. one manifold.

•

••

•E∗

S



Random dispersal in a non-homogeneous habitat.

U,V ecologically equivalent but differ in motility.

Ut = d1∆U + U(m(x)− U − V )

Vt = d2∆V + V (m(x)− U − V ), x ∈ Ω, t > 0

∂νU = ∂νV = 0, x ∈ ∂Ω, t > 0

where habitat suitability m is non-constant and∫

Ω m > 0.

0 < d1 < d2 ⇒ (U,V ) → E1, t → ∞.

Proof: use dependence of principle eigenvalue of L(U) = d∆U + Q(x)U on d to show no coexistence equilibrium

Dockery, Hutson, Mischaikow, Pernarowski, J. Math. Biol. 1998.



Random dispersal versus conditional dispersal

Ut = ∇ · (d1∇U − αU∇m) + U(m(x)− U − V )

Vt = d2∆V + V (m(x)− U − V ), x ∈ Ω, t > 0

∂νV = d1∂νU − αU∂νm = 0, x ∈ ∂Ω, t > 0

α > 0 measures the extent of advection towards favorable habitat.

Results depend on domain geometry.1 U wins if d1 = d2, Ω convex, α small but may not in some

non-convex domains.2 Stable coexistence equilibrium if α large.

Spatial Ecology, Cantrell, Cosner, Lou, CRC,2010; Lam, Ni, Advection-mediated coexistence in general environments,JDE 2014.


Monotone control theory

Monotone control system (MCS)

One has inputs u(t) ∈ U, t ≥ 0, to the dynamical system with statex ∈ X and outputs y ∈ Y :

x ′ = f (x ,u)

y = h(x)

If U,X ,Y have partial order relations ≤Z , Z = U,X ,Y , then thesystem is said to be monotone if

u1 ≤U u2, x1 ≤X x2 ⇒ x(t , x1,u1) ≤X x(t , x2,u2), t ≥ 0

and h : X → Y is a monotone mapping: x1 ≤X x2 ⇒ h(x1) ≤Y h(x2).D. Angeli, E. Sontag, Monotone Control Systems, IEEE Trans. Auto. Control. 2003



Static input-state characteristics

MCSx ′ = f (x ,u), y = h(x)

has a static input-state characteristic kx : U → X if for each constantinput u(t) ≡ u, there exists a globally asymptotically stable equilibriumx = kx (u). Then, one has the input-output characteristic ky = h kx .ky : U → Y is monotone.



Two MCSs with compatible input/output space Y .

x’=f(x,u)y=h(x)u

z’=g(z,y) yv=H(z)



Cascade of MCSs is MCS

x’=f(x,u)y=h(x)u

z’=g(z,y) yv=H(z)



Closed loop systemIf input/output space U have compatible order, the closed loop system is a monotone

x’=f(x,u)y=h(x)u

z’=g(z,y) yu=H(z)



Small Gain Theorem*: scalar inputs/outputs u, y , v

Monotone control systems share input/output ordered space Y :

(1) x ′ = f (x ,u), y = h(x), (2) z′ = g(z, y), v = H(z)

(1) is monotone with U = Y = R having usual order, has outputcharacteristic ky .(2) is monotone with V = R having the opposite order, has outputcharacteristic kv .

If orbits of the cascade system

x ′ = f (x ,H(z)), z′ = g(z,h(x))

are bounded and there is a globally attracting fixed point for

uk+1 = (kv ky )(uk )

then the cascade has a globally asymptotically stable equilibrium.*Angeli, Sontag 2003



Example: control of gene expression

gene expression modeled by the control system

x ′ = β(y − x)

y ′ = u − y , v = αf (x)

where production of mRNA, y , is controlled by u and output v is afunction of the protein product x of the gene.

Static input-state characteristic is kx (u) = (u,u).input-output characteristic kv (u) = αf (u).

Output space V has the usual order if f is increasing and V has theopposite ordering if f is decreasing.



Activator-Inhibitor with 2 genes

activator gene :

x ′

1 = β1(y1 − x1)

y ′

1 = u − y1, v = α1f1(x1)

where f1 > 0 satisfy f ′1 > 0.

inhibitor gene :

x ′

2 = β1(y1 − x1)

y ′

2 = v − y2, u = α2f2(x2)

where f2 > 0 satisfy f ′2 < 0.

closed loop system is not cooperative!



Dynamics of Activator-Inhibitor

Unique equilibrium u = (x1, y1, x2, y2) where g(x2) = x2 andg ≡ α2f2 α1f1 is decreasing map

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x2

g(x2

)

Theorem: If g has no period-two point other than fixed point x2 then uis globally attracting.Proof: small gain theorem


Thank You

Thank You

Monotone Dynamical Systems, H.S. & M. Hirsch, Handbook ofDifferential Equations , Ordinary Differential Equations ( volume2), eds. A.Canada, P.Drabek, A.Fonda, Elsevier, 239-357, 2005.

Monotone Dynamical Systems: an introduction to the theory ofcompetitive and cooperative systems, Amer. Math. Soc. Surveysand Monograghs, 41, 1995.

Collaborators: G. Enciso, M.W. Hirsch, S.-B. Hsu, H.R. Thieme, P.Waltman

NSF Support: Thank You!special session next page


Thank You

SS15: Session on Monotone Dynamical Systems &Applications, J. Mierczynski & S.

Morris W Hirsch, Monotone semidynamical systems with dense periodic pointsMats Gyllenberg, Group actions on monotone skew-product semiflowsYuan Lou, Dispersal in advective environmentsSze-Bi Hsu, Competition for two essential resources with internal storage and periodic inputJianhong Wu, Monotone semiflows with respect to high-rank cones on a Banach spaceKing-Yeung Lam, Persistence Results of a PDE Population Model of Phytoplankton with Ratio DependenceStephen Baigent, Carrying simplicies of continuous and discrete-time systemsFulvio Forni, Open differential positive systems: attractors and interconnectionYi Wang, On heteroclinic cycles of competitive maps via carrying simplices, PS13, 13:30-14:00Xingfu Zou, Asymptotic behaviour, spreading speeds and traveling waves of some dynamical systems, PS13, 14:00-14:30Jian Fang, Bistable Traveling Waves for Monotone Semiflows, PS13, 14:30-15:00Patrick De Leenheer, Persistence of aquatic insect populations subject to flooding, PS13, 15:00-15:30Jifa Jiang, The Decomposition Formula for Stochastic Lotka-Volterra Systems with Identical Intrinsic Growth Rate and itsApplications to Stationary Motions, PS14, 16:00-16:30Wenxian Shen, Criteria for the Existence of Principal Eigenvalue of Time Periodic Cooperative Linear Systems with NonlocalDispersal, PS14, 16:30-17:00

Room: Celebration 6


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Monotone Dynamical Systems: Reflections on New Advances ...halsmith/AIMS.pdf · Applications: consensus in multi-agent networks, ecological systems, in-vivo disease modeling, chemical