Travelling wave solutions to the KPP equation with branching noise arising from initial conditions with compact support

Kliem, Sandra

doi:10.1016/j.spa.2016.06.012

Cited by 5 publications

(35 citation statements)

References 14 publications

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“…(1. 14) We note that the strict positivity of B(θ) follows from (1.14), once B(θ) ≥ 0 is established for all θ > θ c . Our approach relies on establishing the estimate (1.14) along the lines of the corresponding result for contact processes in [3,Lemma 4.2].…”

Section: Introductionmentioning

confidence: 88%

“…Note that υ · = υ · (θ) and u * ,l T +· = u * ,l T +· (θ). From [14,Corollary 4.7 and Notation 1. 3-4.…”

Section: The Terms Under Investigationmentioning

confidence: 99%

“…Note that R 0 (t) = −∞ if and only if τ ≤ t. Extending arguments of Iscoe [11] one can show that R 0 (u(0)) < ∞ implies R 0 (u(t)) < ∞ for all t > 0. In [14,Remark 2.8], C + tem -valued left-and right-upper measures were derived as analogues to the law of ξ (−∞,0]∩Z t , ξ [0,∞)∩Z t , t > 0 and first rough estimates on marker-speeds obtained in Section 4. Let R 0 (t) as in (1.8).…”

Section: Introductionmentioning

confidence: 99%

“…For T > 0, denote by υ T the left-upper measure on C + tem corresponding to L ξ (−∞,0]∩Z T (here, L denotes "law") in the contact process setup. By [14,Remark 2.8], e −2 f,g υ T (df ) = P ½ (−∞,0) (·), u (g) T = 0 , for g ∈ C + tem .…”

Section: Introductionmentioning

confidence: 99%

“…For the remainder, let us recall some notation and Theorem 2.2 from [19] that are often used in the present article. Notation 1.4 (Notation from [19], also see Subsection 1.2 of [14]).…”

Section: Introductionmentioning

confidence: 99%

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Right marker speeds of solutions to the KPP equation with noise

Kliem¹

2019

Ann. Appl. Probab.

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We consider the one-dimensional KPP-equation driven by space-time white noise. We show that for all parameters above the critical value for survival, there exist stochastic wavelike solutions which travel with a deterministic positive linear speed. We further give a sufficient condition on the initial condition of a solution to attain this speed. Our approach is in the spirit of corresponding results for the nearest-neighbor contact process respectively oriented percolation. Here, the main difficulty arises from the moderate size of the parameter and the long range interaction. Stopping times and averaging techniques are used to overcome this difficulty.is established in Tribe [19, Theorem 2.2]. Here, a solution to (1.1) is to be understood in the sense of a weak solution (see Notation 1.4 below). Denote with P u 0 the law of such a solution starting in u 0 ∈ C + tem . By [19, Theorem 2.2], the map f → P f on C + tem is continuous and the family of laws P f , f ∈ C + tem forms a strong Markov family. For ν ∈ P(C + tem ), the space of probability measures on C + tem , denote P ν (A) = C + tem P f (A)ν(df ). Use E u 0 respectively E ν to denote respective expectations.Let τ = inf{t ≥ 0 : u(t, ·) ≡ 0} be the extinction-time of the process. By [16, Theorem 1], there exists a critical value θ c > 0 such that for any initial condition u 0 ∈ C + c \{0} with compact support and θ < θ c , the extinction-time of u solving (1.1) is finite almost surely. For θ > θ c , survival, that is τ = ∞, happens with positive probability.The investigation of the dynamics of solutions to (1.1) is a major challenge, where the main difficulty comes from the competition term −u 2 . Without competition, the underlying additive property facilitates the use of Laplace functionals. Including competition, only subadditivity in the sense of [16, Lemma 2.1.7] respectively Kliem [14, Remark 2.For the process in (1.1) one has a self-duality relationship in the form, v(t) are independent solutions to (1.1) with initial condition u 0 respectively v 0 (cf. [14, (2.1)]). Use P(E) to denote the space of probability measures on E. In [14, Remark 2.5] this self-duality is used to prove existence of a unique upper invariant distribution µ ∈ P(C + tem ) satisfyingfor all T > 0, φ ∈ C + c . In [9, Theorem 1], Horridge and Tribe give sufficient conditions ("uniformly distributed in space") for initial conditions to be in the domain of attraction of µ. They characterize µ by the right hand side of (1.4) and show that it is the unique translation invariant stationary distribution satisfying µ({f : f ≡ 0}) = 1. The result and method of proof are in the spirit of Harris' convergence theorem for additive particle systems (cf. Durrett [5, Theorem 3.3]).Recall the construction of solutions to (1.1) from [17] by means of limits of densities of rescaled long range contact processes. When investigating solutions to the SPDE (1.1), it is only natural to anticipate and/or investigate behavior similar in spirit to the approximating systems. Indeed, [9] successfully applied...

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Section: Introductionmentioning

confidence: 88%

“…Note that υ · = υ · (θ) and u * ,l T +· = u * ,l T +· (θ). From [14,Corollary 4.7 and Notation 1. 3-4.…”

Section: The Terms Under Investigationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…For the remainder, let us recall some notation and Theorem 2.2 from [19] that are often used in the present article. Notation 1.4 (Notation from [19], also see Subsection 1.2 of [14]).…”

Section: Introductionmentioning

confidence: 99%

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Right marker speeds of solutions to the KPP equation with noise

Kliem¹

2019

Ann. Appl. Probab.

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The stochastic Fisher-KPP Equation with seed bank and on/off branching coalescing Brownian motion

Blath

Hammer

Nie

2022

Stoch PDE: Anal Comp

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We introduce a new class of stochastic partial differential equations (SPDEs) with seed bank modeling the spread of a beneficial allele in a spatial population where individuals may switch between an active and a dormant state. Incorporating dormancy and the resulting seed bank leads to a two-type coupled system of equations with migration between both states. We first discuss existence and uniqueness of seed bank SPDEs and provide an equivalent delay representation that allows a clear interpretation of the age structure in the seed bank component. The delay representation will also be crucial in the proofs. Further, we show that the seed bank SPDEs give rise to an interesting class of “on/off”-moment duals. In particular, in the special case of the F-KPP Equation with seed bank, the moment dual is given by an “on/off-branching Brownian motion”. This system differs from a classical branching Brownian motion in the sense that independently for all individuals, motion and branching may be “switched off” for an exponential amount of time after which they get “switched on” again. On/off branching Brownian motion shows qualitatively different behaviour to classical branching Brownian motion and is an interesting object for study in itself. Here, as an application of our duality, we show that the spread of a beneficial allele, which in the classical F-KPP Equation, started from a Heaviside intial condition, evolves as a pulled traveling wave with speed $$\sqrt{2}$$ 2 , is slowed down significantly in the corresponding seed bank F-KPP model. In fact, by computing bounds on the position of the rightmost particle in the dual on/off branching Brownian motion, we obtain an upper bound for the speed of propagation of the beneficial allele given by $$\sqrt{\sqrt{5}-1}\approx 1.111$$ 5 - 1 ≈ 1.111 under unit switching rates. This shows that seed banks will indeed slow down fitness waves and preserve genetic variability, in line with intuitive reasoning from population genetics and ecology.

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Travelling Waves in Monostable and Bistable Stochastic Partial Differential Equations

Kuehn

2019

Jahresber. Dtsch. Math. Ver.

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In this review, we provide a concise summary of several important mathematical results for stochastic travelling waves generated by monostable and bistable reaction-diffusion stochastic partial differential equations (SPDEs). In particular, this survey is intended for readers new to the topic but who have some knowledge in any sub-field of differential equations. The aim is to bridge different backgrounds and to identify the most important common principles and techniques currently applied to the analysis of stochastic travelling wave problems. Monostable and bistable reaction terms are found in prototypical dissipative travelling wave problems, which have already guided the deterministic theory. Hence, we expect that these terms are also crucial in the stochastic setting to understand effects and to develop techniques. The survey also provides an outlook, suggests some open problems, and points out connections to results in physics as well as to other active research directions in SPDEs.

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Travelling wave solutions to the KPP equation with branching noise arising from initial conditions with compact support

Cited by 5 publications

References 14 publications

Right marker speeds of solutions to the KPP equation with noise

Right marker speeds of solutions to the KPP equation with noise

The stochastic Fisher-KPP Equation with seed bank and on/off branching coalescing Brownian motion

Travelling Waves in Monostable and Bistable Stochastic Partial Differential Equations

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