Propagation dynamics of a mutualistic model of mistletoes and birds with nonlocal dispersal

This paper is devoted to the study of the propagation dynamics of a mutualistic model of mistletoes and birds with nonlocal dispersal. By applying the theory of asymptotic speeds of spread and travelling waves for monotone semiﬂows, we establish the existence of the asymptotic spreading speed c ∗ , the existence of travelling wavefronts with the wave speed c ≥ c ∗ and the nonexistence of travelling wavefronts with c < c ∗ . It turns out that the spreading speed coincides with the minimal wave speed of travelling wavefronts. Moreover, some lower and upper bound estimates of the spreading speed c ∗ are provided.


Introduction
In ecology, mutual benefit between different populations is a common phenomenon.A special case is the relationship between mistletoes and birds.Mistletoes are typical aerial stem-parasites plants.Birds eat the fruit of mistletoes to obtain nutrients, energy and water.In turn, mistletoes receive directed movement of their propagules into safe germination sites [3].To better understand the interaction between mistletoes and birds, Wang et al. [26] proposed a reaction-diffusion model where the parameters α, d i , d m , D, d, ω are positive constants, and the time delay τ is non-negative.In this model, u(t, x) and v(t, x) are the densities of mature mistletoes and birds at location x ∈ and time t, respectively, α is the hanging rate of mistletoe fruits to trees, d i and d m are the mortality rates of immature and mature mistletoes, respectively, τ is the maturation time of mistletoes, D is the diffusion rate of birds, d is the conversion rate from mistletoe fruits into bird population.The term v(1 − v) models the logistic growth for bird population which measures the bird population growth due to other food resources besides mistletoes in the habitat, γ ∇(v∇u) is a chemotactic term that models the effect that birds are attracted by trees with more mistletoes, γ is the chemotactic coefficient, and ω is used to reflect the fact that birds may perch on other trees without mistletoes and structures irrelevant to the dynamic process of mistletoes.In [26], the authors studied the spatial pattern formation under two different types of kernel functions k.When = R and γ = 0, Wang et al. [27] further investigated the existence of an asymptotic spreading speed and travelling wave solutions.Note that in (1.1), the Fickian diffusion D v is used to model the random movement of birds.It essentially is a local behaviour and hence maybe not accurate enough to describe the long-range effects of the dispersal of birds.In order to describe the dispersal of birds reasonably, Liang, Weng and Tian [19] introduced a nonlocal operator (Dw)(t, x) = (J * w)(t, x) − w(t, x) = R J(x − y)[w(t, y) − w(t, x)]dy in (1.1) and presented the following nonlocal dispersal model of mistletoes and birds: where x ∈ R and t ≥ 0. In this system, J * v − v models nonlocal dispersal processes of birds; αe −d i τ R k(x − y) u(t−τ ,y) u(t−τ ,y)+ω v(t−τ , y)dy is mature mistletoes recruitment, where the integral with a kernel function k(x − y) expresses the spread of mistletoes fruits by birds from location y to location x and at time t−τ , the Holling type II functional response u u+ω is used to model the fruits removal by birds, and e −d i τ represents the probability of the mistletoe from immature survival to maturity; the term d R k(x − y) u (t,y)  u(t,y)+ω v(t, y)dy represents the growth of birds caused by eating mistletoe fruits; the other terms and parameters have the same meaning as that in (1.1).We should point out that the background and applications of nonlocal dispersal J * v − v are described in Bates et al. [4], Fife [11], Hutson et al. [13], Lee et al. [15], Murray [23] and Medlock and Kot [22].In the past 20 years, nonlocal dispersal equations have been extensively studied.We refer readers to [4,5,7,24,32,34] for travelling wave solutions, [6,14] for asymptotic behaviours of solutions for initial boundary value problems, [8,12,18,33] for spreading speeds and [17,30] for entire solutions.The following hypotheses are imposed in [19]: (H1) Both kernels J(x) and k(x) are non-negative, symmetric and normalised, i.e.
and satisfy ω .It is easy to see that system (1.2) always has a trivial equilibrium E 0 = (0, 0) and a boundary equilibrium E 1 = (0, 1).If (H2) holds, then there exists a unique positive equilibrium where σ = d m αe −d i τ .It was proved in [19] that E 0 and E 1 are linearly unstable with respect to the corresponding kinetic system, while E + is locally asymptotically stable.
It is well known that without birds, the adult mistletoes can only spread in a small area.However, with the nonlocal movements of birds, the mistletoes can invade into new large territories.As such, it is a very interesting problem to model the spatial invasion process of the mistletoes.One way to mathematically characterise this dynamics of the process is travelling wave solution.Travelling wave solutions (in short, travelling waves) of (1.1) are bounded functions with the special form (u(t, x), v(t, x)) = (φ(ξ ), ψ(ξ )), ξ = x + ct, which connect two equilibria E 1 and E + , where c > 0 is the wave speed.Clearly, each wave profile (φ, ψ) to (1.2) satisfies where (φ, ψ)(±∞) = lim ξ →±∞ (φ, ψ)(ξ ).In [19], Liang, Weng and Tian have proved the existence of travelling wave solutions by Schauder's fixed point theorem and upper-lower solutions technique, i.e. there exists c * such that for every c ≥ c * , (1.2) admits a travelling wavefront connecting E 1 and E + .We should remark that the nonexistence of travelling wavefronts c < c * is not addressed in [19].
Another way to characterise the spatial invasion process of the mistletoes into new territories is the spatial invasion speeds (or called asymptotic speeds of spread).The asymptotic speed of spread (in short, spreading speed) was first introduced by Aronson and Weinberger [1] for reaction-diffusion equations and has been an important ecological metric in a wide range of ecological applications, see e.g.[2,20,21] and references therein.Since then, there have been extensive investigations on the spreading speed for various evolution systems, see e.g.[2,9,10,16,20,21,28,31] and references therein.In this paper, we are devoted to investigating the spreading speeds and travelling wavefronts of (1.2).Since system (1.2) is cooperative and its solution maps are monotone, we shall use the theory in [20] to study the existence of spreading speeds for (1.2).Note that the theory of spreading speeds was developed in [20] for monotonic systems under a very general setting.The verification of some abstract assumptions in [20] is highly nontrivial for the solution maps of (1.2) due to the emergence of nonlocal dispersal and time delay along with nonlocal interaction.In addition, we provide the upper and lower bounds of the established spreading speed.
Finally, we investigate the travelling wavefronts of (1.2).With the help of the spreading features, we derive the nonexistence of travelling wavefronts with speed c ∈ (0, c * ).As mentioned earlier, the existence of travelling wavefronts of (1.2) with speed c ≥ c * has been obtained by Liang, Weng and Tian [19] by using Schauder's fixed point theorem together with the upper-lower solutions.However, in order to construct a pair of upper-lower solutions successfully, they needed an additional condition (A) and ω ≥ 1.In this paper, we shall remove these assumptions and prove the existence of travelling wavefronts of (1.2) with speed c ≥ c * .We appeal to the monotone semiflow method which is different from that in [19].Note that the first equation of system (1.2) has no diffusion term and the diffusion term in the second equation is nonlocal dispersal J * v − v. Thus, the solution maps associated with (1.2) are not compact with respect to the compact open topology.Therefore, the theory in [20] is no longer applicable to prove the existence of travelling wavefronts.Fortunately, the monotone semiflow generated by (1.2) has some weak compactness, and hence, we can use the abstract results in [8] to obtain the existence of travelling wavefronts with speed c ≥ c * .Our result shows that the asymptotic speed of spread coincides with the minimal wave speed c * .This paper is organised as follows.In Section 2, we establish the well-posedness and the comparison principle for the initial value problem.In Section 3, we show the existence of the spreading speed of (1.2) and provide some lower and upper bound estimates of the spreading speed.In Section 4, the existence and nonexistence of travelling wavefronts are investigated.

Initial value problem
In this section, we shall investigate the existence and uniqueness theorem of solution to the initial value problem and the comparison theorem.By a change of variables U = u and V = v − 1 in (1.2), we obtain (2.1) The spatially homogeneous system associated with (2.1) is 2) It is easy to see that the equilibria of (1.2), respectively, become E := (0, −1), 0 := (0, 0), K := (u + , v + − 1).
We first study the existence and uniqueness of solution to the initial value problem (2.3).

Lemma 2.1. For any initial value
Proof.Let β > 0.Then, system (2.3) can be rewritten as where It is easy to verify that if we choose β large enough, then F i is nondecreasing in U and V, i = 1, 2. Obviously, system (2.4) is equivalent to the following integral system for t > 0 and x ∈ R. Define the set where (t, x) ∈ R + × R. For any (U, V) ∈ , by the monotonicity of F i , we have and hence, G( ) ⊆ .
For μ > 0 and (U, V) ∈ , we define where and hence, Similarly, one has It then follows that Choose μ > 0 large enough such that 2M β 0 +μ < 1.Then, G is a contracting mapping in .By the contraction mapping theorem, we see that G has a unique fixed point in , which is the solution of (2.3).The proof is complete.
Next, we establish the comparison principle for upper and lower solutions of (2.3).For this purpose, we introduce the definition of upper and lower solutions. ) 3) is defined in a similar way by reversing the inequalities in (2.6).
Lemma 2.3.Let (U, V) and (U, V) be a pair of upper and lower solutions of (2.3).Then, U(t, x) ≥ U(t, x) and V(t, x) ≥ V(t, x) for all t ≥ 0 and x ∈ R.
It then follows that W(t) is a continuous function.We shall prove that W(t) ≥ 0, ∀t ≥ 0. Assume, by contradiction, that the assertion is not true.Then, there exists a number t 0 > 0 such that W(t 0 ) < 0. Since W(t)e −δt with δ > 0 is continuous and W(0) ≥ 0. By the property of continuous function, without loss of generality, for such t 0 , we have Moreover, {x k } ∞ k=1 can be chosen properly as local minimisers of W i (t k , x).Then, we obtain that Hence, we further have (2.7) By the strong maximum principle (see e.g.[17, Theorem 2.1]), we obtain that Next, we show that there exists t 0 ∈ [0, τ ] such that U(t 0 , x) ≡ 0 for all x ∈ R, which means there exists some x such that U(t 0 , x) > 0. Assume, by contradiction, that U(t, x) ≡ 0 for all t and x.It then follows from the first equation in (2.5) that φ 1 (t, x) ≡ 0 for t ∈ [−τ , 0] and x ∈ R, which is a contradiction.Since U t > −d m U, we obtain that for t ∈ [t 0 , t 0 + τ ], U(t, x) ≡ 0 for all x ∈ R. Thus, by the first equation of (2.3), we get Let t 1 (φ) = t 0 + τ .Then by (2.8), we obtain that U(t, x) > 0 for t > t 1 (φ), x ∈ R. The proof is complete.

Existence of spreading speed
In this subsection, we are devoted to establishing that the solution of (2.3) has a spreading speed.
Definition 3.1.A family of mappings {Q t } t≥0 is said to be a semiflow on C K , if the following three properties hold: (i) Q 0 = I, where I is the identity mapping; (ii For any u = (u 1 (θ , x), u 2 (θ , x)) ∈ C, define the reflection operator R by Given y ∈ R, define the translation operator T y by For a given operator Q : C K → C K , we make the following assumptions: (A3) One of the following two properties holds: and there is a positive number ς ≤ τ such that Q[u](θ , x) = u(θ + ς , x) for −τ ≤ θ ≤ −ς , and the operator has the property that https://doi.org/10.1017/S0956792523000311Published online by Cambridge University Press (A5) Q : CK → CK admits exactly two fixed points 0 and K, and for any positive number , there is a Let Q t be the solution map of (2.3), that is, In order to apply the theory in [20] to address the existence of a spreading speed for (2.3), we need to verify that the solution map Q t defined in (3.2) satisfies the above properties (A1)-(A5 is given by for any ψ ∈ Y, where Y is the set of all bounded and continuous functions from R to R, and For any ψ ∈ Y, define • = sup x∈R |ψ(x)|.It is easy to see that a 0 (ψ) = ψ , a 1 (ψ)(x) = R J(x − y)a 0 (ψ)(y)dy ≤ ψ .By induction, we can obtain a k (ψ)(x) ≤ ψ for all k = 0, 1, 2, • • • .By (3.4), we have It is clear that the system (2.3) can be rewritten into the following integral system where For where https://doi.org/10.1017/S0956792523000311Published online by Cambridge University Press Choose t 0 > 0 and for any ε > 0, we let It is easy to see that there exists with Hence, for above ε > 0, choose δ = ε 8 e −σ t 0 such that when φ 1 − φ 2 M (x * ) < δ, by (3.5) and (3.6), we obtain By Gronwall's inequality, we further have This shows that Q t is continuous in φ with respect to compact open topology uniformly for t ∈ [0, t 0 ], which, together with the continuity of Q t in t from Lemma 2.1, implies that Q t is continuous in (t, φ) with respect to the compact open topology.The proof is complete.
By Lemma 3.2, the property (A2) holds.The property (A4) can be guaranteed by Lemma 2.3.It is easy to verify that the property (A5) also holds, see also [27,Lemma 3.7].We just need to prove that the solution map Q t satisfies the property (A3).
Hence, for any ε > 0, there exists δ = ε L , such that for any φ ∈ C K , any x ∈ I, s 1 , for t ≥ τ .On the other hand, if t < τ , we set ς = 1.Then, for the T-invariant set defined in (A3), the set {S 1 , where S 1 is the first component of the operator S defined in (3.1).It is clear that . By the second equation of (2.3), we have By a similar argument as that for Q 1 t , we obtain that t satisfies (A3)(a) for t ≥ τ .In the following, we verify that Q 2 t satisfies (A3)(b) when t ∈ [0, τ ].For any φ ∈ C K , we fix t ∈ (0, τ ] and define We just need to show that for any given compact interval x) for all φ ∈ .Hence, by the precompactness of (0, •) in X, we obtain that S 2 ( ) is equicontinuous on [−τ , −t] × I.
Since P(t) is uniformly continuous for t in a bounded interval in the compact open topology with respect to the initial value, one can show that , where H 2 is defined in (3.7).Then there exists M > 0 such that where the norm Hence, for any ε > 0, there exists δ 1 = min{ ε 2 K 4M , t}, such that for any t ≤ δ 1 , x ∈ I and φ ∈ , we have In [14, Section 2], Ignat and Rossi showed that the solution of (3.3) can also be written as V(t, x) = [P(t)ψ](x) = R G(t, y)ψ(x − y)dy, where G(t, x) = e −Dt δ 0 (x) + R(t, x), δ 0 (x) is the delta measure at zero and R(t, x) = 1 2π R (e D( Ĵ(ξ )−1)t − e −Dt )e jxξ dξ with j = √ −1 and Ĵ being the Fourier transform of J.Moreover, it is proved that |G(t, •)| L 1 (R) ≤ 3 for any t > 0. Since (φ 1 , φ 2 ) ∈ (0, •) and (0, •) is precompact in X, then for the above I, there exists δ 2 > 0 such that for any 12 , and hence, On the other hand, for all t > 0, x ∈ R and φ ∈ C K , we have Hence, for t 1 , t 2 ∈ [0, δ 1 ], (φ 1 , φ 2 ) ∈ (0, •), there exists Finally, we need to verify that S 2 ( ) is equicontinuous on . Thus, we can prove the current case similar to that for (A3)(a).Therefore, S 2 ( ) is equicontinuous on [−τ , 0] × I.The proof is complete.Now we are ready to apply the general theory in [20,Theorem 2.17] to show that the map Q t admits a spreading speed c * , which is also the spreading speed of solutions to (2.3).Theorem 3.4.Assume that (H1) and (H2) hold.Then, there exists a spreading speed c * of Q t in the following sense.(ii) For any c < c * and any σ ∈ CK with σ 0, there exists a positive number r σ such that if φ ∈ C K and φ σ for x on an interval of length 2r σ , then lim t→∞,|x|≤ct U(t, x; φ) = ũ+ and lim t→∞,|x|≤ct V(t, x; φ) = ṽ+ .
be the eigenfunction of the infinitesimal generator corresponding to λ(μ).In fact, ζ can take the form Then, e λ(μ)t is the principle eigenvalue of B t μ with eigenfunction ζ .In particular, γ (μ) := e λ(μ) is the eigenvalue of B 1 μ .Define By using [20, Lemma 3.8], we can easily obtain the following properties of (μ).
Then, we can get an estimate of an upper bound of the spreading speed c * .Proposition 3.6.Let c * be the spreading speed of Q t defined as in Theorem 3.4, and let λ(μ) and (μ) be defined as above.Then, Proof.Clearly, the solution (U(t, x), V(t, x)) of (2.1) is a lower solution of (3.11), and hence, Q 1 (φ) ≤ N 1 (φ) for any φ ∈ C K .It is easy to verify that N 1 and B 1 μ satisfies (C1)-(C6) in [20].By [20,Theorem 3.10], it suffices to show that the principal eigenvalue γ (0) is greater than 1, and the infimum of (μ) is attained at some μ * > 0.
Next, we provide an estimate of lower bound of the spreading speed c * .Proposition 3.7.Let c * be the spreading speed of Q t defined as in Theorem 3.4.Then, Here, (μ) = max{D J(μ) − D − 1, 2 (μ)}, where 2 (μ) is the unique positive root of L( , Proof.Choose any small ε > 0. Let P ε t be the solution operator of the following linear system: where t > 0, x ∈ R. By a similar argument as that in the proof of Proposition 3.6, we can obtain that P ε t satisfies (C1)-(C7) in [20].Moreover, for any given ε ∈ (0, 1), there exists δ = (δ 1 , δ 2 ) such that the solution (U, V) of (3.15) satisfying for any initial φ = (φ 1 , φ 2 ) with 0 ≤ φ 1 ≤ δ 1 , 0 ≤ φ 2 ≤ δ 2 .Hence, (U(t, x; φ), V(t, x; φ)) satisfies and By the comparison principle, we obtain that It then follows from [20,Theorem 3.10] that the spreading speed of P ε t can be attained by the infimum of ε (μ) := ε (μ) μ , where ε (μ) is the principle eigenvalue of which is the characteristic equation for the equation of η corresponding to (3.15).It is easy to verify that the statements in Lemma 3.5 also hold for ε (μ).Then we obtain Since ε > 0 can be chosen arbitrarily, one further has where (μ) is the principal eigenvalue of The proof is complete.
Since the solution map of (2.3) is not compact, we need to use the theory of travelling wavefronts developed in [8] for monotone semiflows with weak compactness to establish the existence of travelling wavefronts of (2.1).Let (X, X + ) be a Banach lattice with the norm • and the positive cone X + .We use M to denote the set of all bounded and nondecreasing functions from R to X and equip M with the compact open topology.We use the Kuratowski measure of noncompactness in X (see e.g.[5]), which is defined by α(B) := inf{r : B has a finite cover of diameter < r} for any bounded set B. It is easy to see that B is precompact (i.e. the closure of B is compact) if and only By employing arguments similar to those in Lemma 2.1, we can easily prove the following wellposedness result.Lemma 4.2.For any initial value φ := (φ 1 , φ 2 ) ∈ M K , system (2.3) has a unique non-negative solution (U(t, x; φ), V(t, x; φ)) which exists globally in time t ≥ −τ , satisfying 0 ≤ (U(t, x; φ), V(t, x; φ)) ≤ K, ∀t ≥ 0. Definition 4.3.A family of mappings {Q t } t≥0 is said to be a semiflow on M β , if the following three properties hold: (i) Q 0 = I, where I is the identity mapping Choose X = R 2 and let Q t be the solution mapping of system (2.3), i.e.Q t = (Q (1)  t , Q (2)  t ) : M K → M K , where (Q (1)  t , Q (2)  t )[φ](θ , x) = (U t (θ , x; φ), V t (θ , x; φ)), (θ , x) ∈ [−τ , 0] × R, t ≥ 0, where φ = (φ 1 , φ 2 ) ∈ M K and (U(t, x; φ), V(t, x; φ)) is the mild solution of system (2.3).
Clearly, the solution mapping {Q t } t≥0 is a semiflow on M K .We need to verify that the solution semiflow Q t satisfies the assumptions in [8] for each t > 0, which are listed as follows.Theorem 4.4.Assume that (H1) and (H2) hold, let c * be the asymptotic spreading speed of Q t defined as in Theorem 3.4.Then, for any c ≥ c * , system (2.1) admits a travelling wavefront (ϕ 1 (x + ct), ϕ 2 (x + ct)) connecting 0 and K. Furthermore, (ϕ 1 (x + ct), ϕ 2 (x + ct)) is also a classical solution to (2.1).

Conclusions
In this paper, we have studied the propagation dynamics of a mutualistic model of mistletoes and birds with nonlocal dispersal.We proved the well-posedness and the comparison principle for the initial value problem.We have also established the existence of the spreading speed and provided the upper and lower bound estimates of the spreading speed.In addition, the travelling wavefronts are considered again.Our result shows that the spreading speed coincides with the minimal wave speed of travelling wavefronts for this model.Our main methods are based on the comparison argument and the theory of asymptotic speeds of spread for the monotone semiflow developed in [8,20].
continuous with respect to the compact open topology.
).It is straightforward to verify that (A1) holds, since (U(t, −x), V(t, −x)) and (U(t, x − y), V(t, x − y)) are also solution of (2.1) provided that (U(t, x), V(t, x)) is a solution (2.1) and y ∈ R. Let Q t be the solution map of (2.3) defined in (3.2).Then, {Q t } t≥0 is a semiflow on C K .Proof.We shall prove that Q t is the continuous in φ with respect to the compact open topology uniformly for t ∈ [0, t 0 ] with t 0 > 0. In view of [29, Lemma 3.1], the solution semigroup of the following linear + ct, c > 0 is the wave speed.Substituting (4.1) into (2.1)gives ⎧ .1) https://doi.org/10.1017/S0956792523000311Published online by Cambridge University Press where ξ = x