Predator-mediated coexistence with chemorepulsionjbell/recent_presentation/AMS17_UNT... · 2017. 9....

Predator-mediated coexistence withchemorepulsion

Jon Bell (UMBC, Emeritus) andEvan Haskell (Nova Southeastern University)

Special Session, AMS, 9-10 September 2017, University ofNorth Texas, Denton

Jon Bell prey-taxis, AMS, 9 September 2017

Model: Two competing prey, a common predator

ut = d4u + u(1− u − a1v − b1w)vt = d4v + v(r(1− v)− a2u − b2w) in Ω× (0,T )wt = ∇ · (∇w + χ0w∇z) + w(−µ+ b1u + b2v)τzt = d14z − µzz + δu

Reduction (“direct interaction”): z = (δ/µz)u

ut = d4u + u(1− u − a1v − b1w)vt = d4v + v(r(1− v)− a2u − b2w) in Ω× (0,T )wt = ∇ · (∇w + χw∇u) + w(−µ+ b1u + b2v)

(1)

Along with homogeneous Neumann b.c.s and initial conditions


Some Theory

Ω ⊂ Rn connected, bounded, ∂Ω smooth, p > n = 2W 1,p

bdy.

= y ∈W 1,p(Ω;R2)|ν · ∇y∂Ω

= 0X

.= y = (u, v ,w) ∈ (W 1,p

bdy )3|y(Ω) ∈ H ; Assume(u0, v0,w0) ∈ X

Theorem: ∃ T = T (u0, v0,w0) s.t. i) there is a unique maximalclassical solution (u, v ,w) on ΩT .

= Ω× (0,T ); ii) (u0, v0,w0) ≥ 0implies (u, v ,w) ≥ 0 on ΩT ; iii) if ||(u(·, t), v(·, t),w(·, t))||L∞ isbounded away from boundary of H for all t ∈ (0,T ), thenT = +∞: i.e. (u, v ,w) is global in time.

Proof similar to arguments in Wrzosek, 2004, Wang and Hillen,2007, and relies on Amann’s theory, 1990, 1993.


Constant Steady State (us , vs ,ws)

Non-spatial (ODE) case: For competitive system (no predator),a1 > 1, a2 < r , v “wins”, u “loses” ((u, v)→ (0, 1) as t →∞)a1 < 1, a2 > r , u “wins”, v “loses” ((u, v)→ (1, 0) as t →∞).

Full ODE system: there is a non-empty (admissible parameter set)A such that

Proposition (predator-mediated coexistence):(a1, a2, b1, b2, r , µ) ∈ A, then (us , vs ,ws) ∈ (R+)3 and is globallystable positive solution.

Assume parameters always in A.

Question: Can a (chemotactic) prey defense destabilize thesituation? Is there pattern formation? How do the two cases abovediffer?


Convergence to the Constant Steady State

Let (a1, a2, b1, b2, r , µ) ∈ A.

V =∫ uus

u′−usu′ du′ + v ′−vs

v ′ dv ′ + w ′−wsw ′ dw ′

V (t) = V (u, v ,w) =∫

Ω V (u, v ,w)dxHL = (u, v ,w)|V (u, v ,w) ≤ L

L can be large enough for HL to be positive invariant.

Proposition: Let (u0, v0,w0) ∈ HL. If r > (a1 + a2)2/4 andχ2 < 4dus/(wsu

2max), then V (t)→ 0 as t →∞; i.e., (us , vs ,ws) is

globally asymptotically stable for system (1).

Proof: V (t) is a strong Lyapunov functional


Pattern Formation

Setup: linearize about the constant steady state

∂

∂t

U

V

W

=

d 0 0

0 d 0

χws 0 1

4

U

V

W

+

−us −a1us −b1us

−a2vs −rvs −b2vs

b1ws b2ws 0

U

V

W

Consider (U,V ,W ) = eλt+i~k·~x ~ψ, k = |~k |:

J =

dk2 + us + λ a1us b1us

a2vs dk2 + rvs + λ b2vs

χwsk2 − b1ws −b2ws k2 + λ


Dispersion Relation

The det(J ) = 0 gives the dispersion relation (DR):

λ3 + a(k2)λ2 + b(k2, χ)λ+ c(k2, χ) = 0

DR gives relation between wave number k and temporal behavior(frequency ∼ 1/λ)

instability if either c(k2, χ) < 0 or a(k2)b(k2, χ)− c(k2, χ) < 0

Under some technical conditions on parameters, there exist aχc(k2, d) > 0 such that

c(k2, χ) < 0 iff χ > χc(k2, d)


Chemotaxis-induced instability

χ > χc does not guarantee pattern formation Ω is bounded.

Figure: A plot of χ versus k2. The blue graph is χc as a function of k2.

Assume (a1, a2, b1, b2, r , µ) ∈ A, b1dk2 > vs(a1b2 − rb1)

Theorem: Let Ω = (0, L). Let χm > χc be the first value of χsuch that either k1L/π or k2L/π is an integer. Then χm is abifurcation parameter and χ > χm is a necessary and sufficientcondition for pattern formation of system (1).


Diffusion-induced instability

c(k2, χ) can be written in form

c(k2, χ) = k6d2 + A(k2, χ)d + B(k2, χ)

A(k2, χ) = b1uswsk4χc1(k2, r)− χ

B(k2, χ) = usvsws(rb1 − a1b2)k2χc2(k2, r)− χ

χc1(k2, r) is always positive, decreasing in k2 to a positive limit.

χc2(k2, r) is dependent on signs of rb1 − a1b2, r − a1a2, etc.

Example: rb1 − a1b2 > 0, r − a1a2 > 0. Then χc2 > 0, decreasingto a positive limit. Assume parameters s.t. there exists k2 > 0(χc2 > χc1 iff k2 > k2), and k2 > 0 large enough so (0, k2)contains a mode


Diffusion-induced instability continued

Figure: A plot of χc1 (red), χc2 (green) versus k2.

(k2, χ) in I, II: (us , vs ,ws) unstable for 0 < d < d+;in III: (us , vs ,ws) unstable for 0 < d− < d < d+;in IV: (us , vs ,ws) always locally stable


Case 1: u wins


Case 2: v wins


Case 2: Early epoch time vs steady state


Summary

I Consistency across finite difference, continuous FE,discontinuous FE methods, and ranges of parameters

I Here graphs with “slow” prey (d = 0.1); for d ≈ 1, predator“bumps” are wider

I Prey are “out of phase” with predator, v tends to hide behindu

I “Win, don’t loose” dynamics


Thank you for your attention

Figure: This is AfterMath, my cruising home


Predator-mediated coexistence with chemorepulsionjbell/recent_presentation/AMS17_UNT... · 2017. 9....

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Transcript of Predator-mediated coexistence with chemorepulsionjbell/recent_presentation/AMS17_UNT... · 2017. 9....