Persistence theory applied to Keen’s model– a link between mathematical biology and mathematical

economics

Jianhong Wu and Xiang-Sheng Wang

Mprime Centre for Disease ModellingYork University, Toronto

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Outline

• Introduction on persistence theory

• Keen’s model without government intervention

• Keen’s model with government intervention I

• Keen’s model with government intervention II

• Discussion on uniformly strongly persistence and sustained oscillations

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Persistence theory on mathematical biology

H. Smith and H. R. Thieme, Dynamical Systems and Population Persistence,Graduate Studies in Mathematics, 118. American Mathematical Society,Providence, RI, 2011.

• Which species, in a mathematical model of interacting species, will survive overthe long term?

• In a mathematical model of an epidemic, will the disease drive a hostpopulation to extinction or will the host persist?

• Can a disease remain endemic in a population?

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Persistence definitions

Let Φ(t, x) : R+×X → X be the semiflow generated by a differential system withinitial values x ∈ X. For a nonnegative functional ρ from X to R+, we say

• Φ is ρ - uniformly strongly persistent (USP) if lim inft→∞ ρ(Φ(t, x)) > ε forany x ∈ X with ρ(x) > 0.

• Φ is ρ - uniformly weakly persistent (UWP) if lim supt→∞ ρ(Φ(t, x)) > ε forany x ∈ X with ρ(x) > 0.

• Φ is ρ - strongly persistent (SP) if lim inft→∞ ρ(Φ(t, x)) > 0 for any x ∈ Xwith ρ(x) > 0.

• Φ is ρ - weakly persistent (WP) if lim supt→∞ ρ(Φ(t, x)) > 0 for any x ∈ Xwith ρ(x) > 0.

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Example: Goodwin’s model

• (Goodwin 1967) Predator-prey system of wage share (ω) and employment rate(λ):

ω′ = ω[Φ(λ)− α];

λ′ = λ[π/ν − α− β − δ],

where π = 1− ω is the profit share.

• Lyapunov functional

V (ω, λ) =

∫ λ Φ(s)− αs

ds−∫ ω (1− s)/ν − α− β − δ

sds.

• The Goodwin’s model is eπ - UWP and eπ - SP but not eπ - USP.

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Example: Goodwin’s model

The Goodwin’s model is eπ - UWP and eπ - SP but not eπ - USP.

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Notations

• Variables: ω is the wage share, λ is the employment rate, d is the debt ofcapitalists, gS is the government spending, gT is the tax share, and π is theprofit share.

• Parameters: α is the growth rate of productivity, β is the growth rate of totallabor force, δ is the depreciation rate in capital, ν is the capital-to-output ratio,and r is the interest rate.

• Functions: Φ(λ) is the Phillips curve, κ(π) is the investment function, η(λ) isthe government spending function, and Ξ(π) is the tax function.

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Keen’s model without government intervention

• (Keen 1995) Three-dimensional system of wage share (ω), employment rate (λ)and capital debt (d):

ω′ = ω[Φ(λ)− α];

λ′ = λ[κ(π)/ν − α− β − δ]; (1)

d′ = [κ(π)− π]− d[κ(π)/ν − δ],

where π = 1− ω − rd is the profit share.

• ω = 0, λ = 0 and d =∞ is locally asymptotically stable if κ(−∞)/ν − δ < rand r > 0.

• ω = 0, λ = 0 and d = −∞ is locally asymptotically stable if κ(−∞)/ν − δ < rand r < 0.

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Persistence results

• If κ(−∞)/ν − δ > r, then (1) is ed - USP and eπ - UWP:

lim inft→∞

d(t) > −M and lim supt→∞

π(t) > −M.

• If κ(−∞)/ν − δ > r and r > 0, then (1) is e−π - USP and e−d - UWP:

lim supt→∞

π(t) < M and lim inft→∞

d(t) < M.

• If r 6= 0, then (1) is e−π - UWP: lim inft→∞ π(t) < M.

• If r = 0, then (1) is e−π - USP, eπ - UWP, eπ - SP, e−d - UWP and ed - UWP.

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Keen’s model with government intervention I

(Keen 1995) Five-dimensional system of wage share (ω), employment rate (λ),capital debt (d), government spending (gS) and tax (gT ):

ω′ = ω[Φ(λ)− α];

λ′ = λ[κ(π)/ν − α− β − δ];d′ = [κ(π)− π]− d[κ(π)/ν − δ]; (2)

g′S = η(λ)− gS[κ(π)/ν − d];

g′T = Ξ(π)− gT [κ(π)/ν − d],

where π = 1−ω− rd+ gS − gT is the profit share. Let u = rd− gS + gT , we have

u′ = r[κ(π)− π]− u[κ(π)/ν − δ]− η(λ) + Ξ(π).

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Keen’s model with government intervention I

• The system (2) can be reduced to a three-dimensional system with wage share(ω), employment rate (λ) and normalized debt (u):

ω′ = ω[Φ(λ)− α];

λ′ = λ[κ(π)/ν − α− β − δ]; (3)

u′ = r[κ(π)− π]− u[κ(π)/ν − δ]− η(λ) + Ξ(π),

where π = 1− ω − u is the profit share.

• ω = 0, λ = 0 and u =∞ is locally asymptotically stable if κ(−∞)/ν − δ < r.

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Persistence results

• If κ(−∞)/ν − δ > r and r ≥ 0, then (3) is eu - USP, e−π - USP, eπ - UWPand e−u - UWP:

−M < lim supt→∞

π(t) < M and −M < lim inft→∞

u(t) < M.

• If κ(−∞)/ν − δ > r and r < 0, then (3) is e−u - USP, e−π - UWP and eπ -UWP:

lim supt→∞

u(t) < M, lim inft→∞

π(t) < M and lim supt→∞

π(t) > −M.

• (3) is always e−π - UWP: lim inft→∞ π(t) < M.

• If η(λ) = O(1/λ) as λ→ 0 and α+ β > r, then (3) is eπ - UWP:lim supt→∞ π(t) > −M.

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Keen’s model with government intervention II

(Keen 1995, modified by Grasselli and Costa Lima 2012) Five-dimensional systemof wage share (ω), employment rate (λ), capital debt (d), government spending(gS) and tax (gT ):

ω′ = ω[Φ(λ)− α];

λ′ = λ[κ(π)/ν − α− β − δ];d′ = [κ(π)− π]− d[κ(π)/ν − δ]; (4)

g′S = gS{η(λ)− [κ(π)/ν − d]};g′T = gT{Ξ(π)− [κ(π)/ν − d]},

where π = 1− ω − rd+ gS − gT is the profit share. We assumeΞ(−∞) < κ(−∞)/ν − δ.

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Persistence results

• ω = 0, λ = 0, d =∞, gS = 0 and gT = 0 is locally asymptotically stable ifη(0) < κ(−∞)/ν − δ < r.

• ω = 0, λ = 0, d =∞, gS =∞ (with gS � d) and gT = 0 is locallyasymptotically stable if κ(−∞)/ν − δ < η(0) < r.

• (4) is always e−π - UWP: lim inft→∞ π(t) < M.

• If κ(−∞)/ν − δ > r and r ≥ 0, then (4) is e−π - USP: lim supt→∞ π(t) < M.

• If κ(−∞)/ν − δ > r, then (4) is eπ - UWP: lim supt→∞ π(t) > −M.

• If η(0) > r, then (4) is eπ - UWP: lim supt→∞ π(t) > −M.

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Brief summary

• Keen’s model is always e−π - UWP: lim inft→∞ π(t) < M.

• If κ(−∞)/ν − δ > r and r ≥ 0, then Keen’s model is e−π - USP:lim supt→∞ π(t) < M.

• If κ(−∞)/ν − δ > r, then Keen’s model is eπ - UWP: lim supt→∞ π(t) > −M.

• For the Keen’s model with government intervention I, if η(λ) = O(1/λ) asλ→ 0 and α+ β > r, then it is still eπ - UWP: lim supt→∞ π(t) > −M.

• For the Keen’s model with government intervention II, if η(0) > r, then it isstill eπ - UWP: lim supt→∞ π(t) > −M.

• Question: when do we have eπ - USP: lim inft→∞ π(t) > −M?

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Discussion: from UWP to USP

• The Goodwin’s model is eπ - UWP and eπ - SP but not eπ - USP.

ω′ = ω[Φ(λ)− α];

λ′ = λ[π/ν − α− β − δ],

where π = 1− ω is the profit share.

• We will have eπ - USP by introducing self-adjustment for the wage share andemployment rate:

ω′ = ω[Φ(λ)− α−A1(ω)];

λ′ = λ[π/ν − α− β − δ −A2(λ)].

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Discussion: from UWP to USP

• For the Keen’s 3D model with self-adjustment, we also have eπ - USP providedκ(−∞)/ν − δ > r and r ≥ 0.

ω′ = ω[Φ(λ)− α−A1(ω)];

λ′ = λ[κ(π)/ν − α− β − δ −A2(λ)];

d′ = [κ(π)− π]− d[κ(π)/ν − δ].

• The proof is organized as follows:

1. We first show that π is eventually uniformly bounded above.2. Next, we prove by using self-adjustment terms that ω and λ are also

eventually uniformly bounded above.3. Finally, we have π eventually uniformly bounded below, namely, eπ - USP.

• Remark: it is strange but most likely the case that in the proof of USP werequire eventually uniform boundedness (certain compactness) of the semiflow.

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Discussion: sustained periodic orbits

Keen’s 3D model without government intervention:

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Discussion: sustained periodic orbits

Keen’s 5D model with government intervention I:

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Discussion: sustained periodic orbits

Keen’s 5D model with government intervention II:

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Thank you!

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