A time-space periodic population growth model with impulsive birth

This paper is devoted to the study of spatial dynamics for a time-space periodic population growth model with impulsive birth. We ﬁrst formulate a discrete-time semiﬂow by the one-year solution map, and obtain a threshold-type result for the semiﬂow with the spatially periodic initial data. Then we establish the existence and a computational formula of the spreading speed and prove the coincidence of the spreading speed with the minimal speed of spatially periodic traveling waves in the monotone case. Further, we investigate the global dynamics of this model in a bounded spatial domain. Finally, we conduct numerical simulations to verify our analytic results and illustrate some interesting ﬁndings.


Introduction
Reproductive synchrony refers to the temporal clustering of reproductive events among individuals within a restricted time, such as mating, spawning, and births, which is widely recorded in plant and animal populations (see [18,19]).It may arise simply as an adaptive mechanism of environmental seasonality in climate or resources and the seasonal growth of plant and animal populations.Some modeling studies have been performed to understand the mechanisms of reproductive synchrony and its effect on population growth.To catch the every-other-day egg laying synchrony pattern observed in colonies of gulls, two difference equation models with juvenile-adult structure were proposed and analyzed in [20,21].Such disturbances typically occur for a short time; nevertheless, they significantly impact the population density and number.Classical differential equations may not well describe the phenomena where the critical drivers are non-continuous processes.A well-accepted modeling approach that includes the reproductive synchrony is to divide the cycle into two seasons, a nonreproduction season and a short reproduction season.Recently, impulsive differential equations have been introduced to model and characterize these hybrid discrete-continuous processes, see, e.g., [1,11,23,25,26].
Suppose the duration of the cycle is fixed to be one year.In that case, the differential system with the impulsive reproduction rate may define an abstract difference equation or a discrete-time semiflow, which describes the evolution of offspring density from this year to the following year [10,25].In this vein, a reaction-diffusion equation model with impulsive seasonal reproduction and individual dispersal was proposed in [5], and an impulsive integro-differential model was analyzed in [25] to capture the dynamics of an invading species with an impulsive reproduction stage and a nonlocal dispersal stage.Further, the problem of persistence and extinction was addressed in [23] for moving animal species with birth pulse and habitat shift.
More recently, the propagation dynamics of species growth with annually synchronous emergence of adults was studied via an impulsive reaction-diffusion model in [1], where the critical domain size was also determined for the persistence of species in a bounded spatial domain with a lethal exterior.
To take into account of climate changes and seasonality, it is more reasonable to incorporate the temporal and spatial variations into these impulsive models.Periodicity is one of the simplest environmental heterogeneities and is a good candidate for approximating complex heterogeneity.There are also similar situations in the ecological environment.For example, a theoretical river may be composed of torrents and slow flows so that the related parameters can be assumed to be spatially periodic.Another example is that in continuous mountains, the living environment of species is closely related to the altitude and light, which make some parameters change periodically between mountains.The purpose of the current paper is to incorporate the spatio-temporal periodic variability into the model proposed in [1] and study the global dynamics of the resulting model by appealing to the theory of monotone evolution systems and the comparison arguments.In the case of an unbounded spatial domain, we obtain sufficient and necessary conditions for the monostable structure of the associated timespace periodic system with spatially periodic initial data, which is different from the spatially homogeneous system discussed in [1].Further, we need to overcome certain difficulties induced by the spatio-temporal periodicity to derive a computational formula of the spreading speed.
In the case of a bounded spatial domain, we choose to use the fractional power space to deal with the Neumann, Robin type, and Dirichlet boundary conditions in a unified way.Our numerical simulations also give rise to some interesting findings in the presence of spatiotemporal periodicity.
The rest of the paper is organized as follows.In section 2, we formulate a time-space periodic reaction-diffusion model with an annual impulsive maturation emergence term to describe the spatial evolution of adult population density, and show that solutions of this model generate a discrete-time semiflow.In section 3, we obtain a threshold-type result for the semiflow with the spatially periodic initial data, establish the existence and a computational formula of the spreading speed in both monotone and non-monotone cases, and further prove that in the monotone case, the spreading speed coincides with the minimal speed of spatially periodic traveling waves.In section 4, we study the global dynamics of the model in a bounded spatial domain.Numerical simulations are conducted to interpret the obtained analytical results, and a brief discussion concludes the paper.
Assume that the immature individuals are reproduced at density g(x, N n (x)) at the beginning of n-year and location x.Hence, v n (0, x) = g(x, N n (x)).The birth function g(x, N ) can be defined as g 1 (x, N ) = s(x) pN q+N and g 2 (x, N ) = s(x)N e r−kN with a spatial periodic function s(x), positive constants p, q, r and k, where pN q+N and N e r−kN are the Beverton-Holt function and the Ricker function, respectively.Note that functions g 1 and g 2 are monotone and nonmonotone with respect to N , respectively.Assume that these immature individuals develop into the adult stage after time τ for τ ∈ (0, 1].The synchronized maturation in the n-th year, the adult population density u n (t, x) has an abrupt increase at time t = τ and location x, that is, where u n (t, x) = u n (t − , x) and R(x; N n ) describes the density of newly emerging matured individuals at time t in the n-th year.Assume that the dispersal is symmetric and follows Fick's diffusion law.Then the evolution of the adult u n (t, x) is governed by on other time instances of one year, where D M (t, x) > 0 is the random diffusion rate and f (t, x, u) is the death rete function.Also, the evolution of the immature v n (t, x) is governed by where D I (t, x) > 0 and d I (t, x) > 0 are the random diffusion rate and the natural death rate, respectively.Let T (t, s), t ⩾ s, be the evolution family on BC(R, R) associated with the linear reaction-diffusion equation This implies that the density of newly emerging matured individuals at location x and time It then follows that we have the following time-space periodic evolution system on the adult population density with an annual impulsive maturation emergence: (2.1) Figure 1: The schematic diagram of the evolution of population dynamics for the adult un(t, x) and juvenile vn(t, x) at time n + t and location x, here t ∈ [0, 1] and n represents the n-th year.
with the initial data u 0 (0, x) = N 0 (x), ∀x ∈ R. The schematic diagram of the evolution of the adult and juvenile population is described in Figure 1.We further assume that (H1) All functions D M (t, x), D I (t, x) and d I (t, x) are in C ν 2 ,ν (R + × R) for some ν ∈ (0, 1), and 1-periodic in t and L-periodic in x for some L > 0, and there exists a number α > 0 such that D i (t, x) ⩾ α, i = M, I, for all (t, x) ∈ R + × R.
x, 0), and f (t,x,u) u is strictly decreasing in u.
It follows from (H2)-(H3) that there exists a positive number N such that R(x; N ) + f (τ, x, N ) ≤ 0 for all x ∈ R. Let M (t, s), t ⩾ s, be the evolution family on BC(R, R) associated with ∂u ∂t = ∂ ∂x (D M (t, x) ∂u ∂x ) + f (t, x, u).For adult density φ n (x) at location x at the beginning of the n-th year, the distribution at time τ in the same year is [M (τ, 0)φ n ] (x), and at time τ After that, the population continues to evolve from time τ and to time 1, and the distribution at the end of the year becomes Therefore, the time-one solution map of system (2.1) is For each time instant t ⩾ 0, there is a unique decomposition t = t + t, where t ∈ [0, 1) and t denotes the nearest integer less than or equal to t.Thus, the time-t solution map of system (2.1) can be expressed as with the initial data φ(x) = N 0 (x), ∀x ∈ R. Clearly, Φ Accordingly, we will focus on the evolution dynamics of the discrete-time semiflow {Q n } n∈N associated with system (2.3) in the unbounded and bounded domains, respectively.

Spreading speeds and traveling waves
In this section, we first study the spreading speed and the spatially L-periodic traveling waves of system (2.3) in the monotone case, and then investigate the spreading speed of this system in the non-monotone case by a comparison method.
3.1 The monotone case of g(x, N n ) We first present the threshold dynamics of system (2.3) with spatially periodic initial conditions, and then investigate the spreading speeds and spatially periodic traveling waves.
Let E per be the set of all continuous and L-periodic functions from R to R with the maximum norm ∥ • ∥ E per , and a strongly ordered Banach lattice.We consider the following discrete-time system associated with (2.3): where φ(x + L) = φ(x), ∀x ∈ R. It follows from the conditions (H1)-(H3) and the monotonicity of function g that Q is monotone and strongly subhomogeneous (see [27]).By the definition of Q and the chain rule, we can compute the Fréchet derivative DQ(0) on E per as follows where M (t, s), t ⩾ s, is the evolution family on E per associated with the linear reaction-diffusion equation It then follows that the linear discrete-time recursion is that is, Clearly, Q is compact in E per + , and DQ(0) is compact and strongly positive in E per .Let r(DQ(0)) be the spectral radius of DQ(0).As a straightforward consequence of [ (ii) If r(DQ(0)) > 1, then system (2.3) admits a unique positive L-periodic steady state , and it is globally asymptotically stable in E per + \{0}.
In order to study propagation dynamics for system (2.3), we assume that r(DQ(0)) > 1 in the rest of this subsection so that the operator Q admits a globally stable positive L-periodic Let C be the set of all bounded and continuous functions from Theorem 3.2.Assume that (H1)-(H3) hold and r(DQ(0)) > 1.Then the following statements are valid: Proof.We appeal to the theory developed by [13] to establish the existence of the asymptotic speed of spread for (2.3) under assumptions (H1)-(H3) and r(DQ(0)) > 1.Under these assumptions, we see from Proposition 3.1 that N * is the L-periodic steady state of (2.3).For for any y ∈ R and any function h : R → R, we define the translation operator Ty by We claim that Q is space-L periodic in the sense that Q • Ty (φ) = Ty • Q(φ) for all y ∈ LZ, φ ∈ C N * .Indeed, for any y ∈ LZ, u( t, x; Ty (φ)) and u( t, x − y; φ) are solutions of (2.1) with initial conditions u(0, x; Ty (φ)) = Ty (φ)(x) = φ(x − y) and u(0, x − y; φ) = φ(x − y), respectively.
It then follows from the uniqueness of solutions of (2.1) that the claim holds true.Using the notations there, we set follows from above analysis that Q satisfies (E1), (E2) and (E4) in [13].Note that Q is compact on C N * , and hence (E3) holds.Under the condition r(DQ(0)) > 1, we see from Proposition To obtain a computational formulas of c * ± , we use the linear operators approach (see [12,24]).
Denote R 1 ( t) as the solution map of system (3.5) on C per .Then where w( t, x; σ) is the solution of (3.4), and hence, Let Λ(µ) = ln r(P µ ).Then σ µ is a positive fixed point of R 1 (1), and hence, the solution µ .By Lemma 3.3, it follows that for any given µ > 0, there exists the corresponding time-space periodic principal eigenfunction r 1 ( t, x) > 0 with the principal eigenvalue Λ(µ) satisfying (3.5).Due to the time-space periodicity of r 1 ( t, x), for any given µ > 0, there is a ( tµ , x µ ) ∈ R 2 such that r 1 ( tµ , x µ ) = min ( t,x)∈R 2 r 1 ( t, x), which implies that Letting ( t, x) = ( tµ , x µ ) in (3.5), and the unique decomposition tµ = n + t µ , we see from (3.6) that and hence, This, together with the boundedness of ∂ x D M (t µ , x µ ) and ∂ un f (t µ , x µ , 0), implies that Since Λ(0) > 0, we have lim By virtue of (3.7) and (3.8), it is easy to see that Φ(µ) attains its minimum at some finite µ * .Since the solution of (2.3) is a lower solution of the linear system (3.3)under the conditions (H2) and (H3), we have By the arguments similar to those for [24, Theorem 2.5] and [12, Theorem 3.10(i)], we obtain For any given ε ∈ (0, 1), there exists δ = δ(ε) such that . By the comparison principle, we have It then follows that for any ρ ∈ C ξ , the solution u( t, x; ρ) of (2.3) satisfies Consider the linear system Let {L ε (t, s) : t > s} be the evolution family on C generated by the above linear system.Furthermore, the comparison principle implies that and hence, For µ ≥ 0, let {L ε µ (t, s) : t > s} be the evolution family on C generated by the following system where Rε (x; where r(L ε µ (1, 0)) is the spectral radius of the Poincaré map associated with system (3.9).
By the analysis on L ε µ ( t, 0) similar to those for L µ ( t, 0) and the arguments similar to those For any x ∈ R, define Clearly, g + is nondecreasing with respect to N n , ∂ Nn g + (•, 0) = ∂ Nn g(•, 0), and g + satisfies the assumption (H3) for g.In the case r(DQ(0)) > 1, Proposition 3.1(ii) implies that (2.3) with g replaced by g + has a positive L-periodic steady state N * + (x).Then for any x ∈ R, define Clearly, g − is nondecreasing with respect to N n , ∂ Nn g − (•, 0) = ∂ Nn g(•, 0), and g − also satisfies the assumption (H3).Similarly, system (2.3) with g replaced by g − admits a positive Lperiodic steady state N * − (x).Note that 0 We now consider two auxiliary systems: and where . Similar to the procedures of recursion operator Q in (2.2), systems (3.10) and (3.11) define two recurrence relations for N ± n+1 (x) as and respectively.Let N + n (x), N − n (x) and N n (x) be the solutions of (3.12), (3.13) and (2.3), respectively.Thus, the comparison arguments imply that if Note that c * ± in subsection 3.1 are the spreading speeds of (2.

Global dynamics in a bounded domain
In this section, we study the evolution dynamics of (2.1) in a bounded domain Ω under the boundary conditions where Ω ⊂ R n , (n ⩾ 1), and if n > 1, we suppose that ∂Ω is a class of C 2+θ (0 < θ ≤ 1).Further ,ν (R + × Ω) for some ν ∈ (0, 1) and 1-periodic in t, and there exists a number α > 0 such that D i (t, x) ⩾ α, i = M, I, for all u is strictly decreasing in u.
Let p ∈ (n, ∞) be fixed, and X := L p (Ω).For each γ ∈ 1 2 + n 2p , 1 , let X γ be the fractional power space of X with respect to Theorem 1.11] that system (2.1) generates a local time-one periodic semiflow Φt t⩾0 on X γ in a weak sense, and bounded orbits in X γ are precompact.In view of (A2)-(A3), there exists a number N > 0 such as By the L ∞ -boundedness of solutions of system (2.1), we obtain the global existence of solutions.
Similar to the arguments for the discrete-time recursion (2.3) with the bounded domain, we can investigate the following recursion Clearly, the time-one solution map Q := Φ1 satisfies where T (t, s), M (t, s), t ⩾ s, are the evolution family on X γ associated with the linear equations x, u) with the boundary conditions Bv = 0 and Bu = 0, respectively.Moreover, the map Q is compact in X γ .
Next, we focus on the evolution dynamics of the discrete-time semiflow Qn n∈N associated with system (4.1) in the monotone and non-monotone cases, respectively.
4.1 The monotone case of g(x, N n ) We consider system (4.1)under assumptions (A1)-(A3) and the monotonicity of the birth function g.Clearly, Q is monotone in X γ .With (A2)-(A3), we can easily show that Q is strongly subhomogeneous in X γ .
By the definition of Q and the Chain rule, we compute the Fréchet derivative D Q(0) on X γ as follows where M l (t, s), t ⩾ s, is the evolution family on X γ associated with the linear reaction-diffusion equation Then we have the following linear discrete-time recursion that is, Denote the spectral radius of D Q(0) as r(D Q(0)).Then we have the following threshold-type result.
Theorem 4.1.Assume that (A1)-(A3) hold, and the birth function g is monotonically increasing, and let N n (x; φ) be the solution of (4.1) with φ ∈ X γ + .Then the following statements are valid: (i) If r(D Q(0)) < 1, then the zero steady state is globally asymptotically stable in X γ + .
Therefore, we have r(D Q(N * )) < 1, and hence, N * is linearly asymptotically stable.This, together with the global attractivity of N * , implies that N * is globally asymptotically stable in X γ + \{0}.

4.2
The non-monotone case of g(x, N n ) Now we consider system (4.1)under assumptions (A1)-(A3) and non-monotonicity of the birth function g(x, N n ).Clearly, Q is not monotone in X γ .In this subsection, we need the following additional assumption: (A4) There is a > 0 such that for each x ∈ Ω, g(x, N n ) is nondecreasing with respect to For any x ∈ Ω, we define g + (x, N n ) and g − (x, N n ) as in subsection 3.2.Then we have the following two auxiliary systems: and where R± (x; . Thus, the time-one solution maps of systems (4.5) and (4.6) give rise to the following discretetime recursions: and Using the same arguments as in subsection 4.1, we see that systems (4.5) and (4.6), with the monotone birth functions g + and g − respectively, admit the same linearized system, and the spectral radii of the time-one map of this linearized system is r(D Q(0)).Using (4.9) and the comparison arguments, we have the following result.

Numerical simulations
In this section, we present some numerical simulations for system (2.1).We first investigate the influence of τ for the evolution of adult population.For simplicity, let the loss function Both of these situations can occur in ecology.In the first case, for example, some species will move into the wild after they mature, which makes them easy to be hunted, that is,

Discussion
In this paper, by incorporating the temporal and spatial variations into an impulsive system, we propose a time-space periodic reaction-diffusion model with an annual impulsive maturation emergence term to study the invasion dynamics in unbounded and bounded domains, respectively.We reduce the study of the evolution dynamics of such a model to that of a discrete-time system from the evolution viewpoint.When the habitat is unbounded, we obtain the existence of the spreading speeds in both monotone and non-monotone cases and show they are linearly determinate.We further prove that the spreading speeds in the monotone case coincide with the minimal speeds of spatially periodic traveling waves.
When the habitat is bounded, we introduce the fractional power space to deal with general In particular, we give basic ecological reasons for these two numerical results.We also use numerical experiments to illustrate the long-time behaviour of solutions of system (2.1) with time-space periodic parameters, which indicates that the solutions of system (2.1) connect 0 and time-space periodic solutions in the monotone and non-monotone cases, respectively.
It is worth mentioning that in the non-monotonic case, the nonexistence of spatially periodic traveling waves is a straightforward consequence of the spreading speeds (see, e.g., [25, Theorem 3.5(i)]).However, the existence of spatially periodic traveling waves in this case is still an challenge open problem.We leave it for future investigation.
where |•| denotes the usual norm in R. Then (C, ∥•∥ C ) is a normed space.Let d(•, •) be the distance induced by the norm ∥•∥ C .It follows that the topology in the metric space (C, d) is the same as the compact open topology in C, that is, a sequence of points φ n converges to φ in C if the sequence φ n (x) of functions converges to φ(x) uniformly for x in any compact subset of R. Assume that β is a strongly positive L-periodic continuous function from R to R. Define ii) If φ ∈ C N * and φ ̸ ≡ 0, then for any c and c satisfying −c * − < −c < c < c * + , there holds lim n→∞,−cn⩽x⩽cn

3. 1
that (E5) holds.It then follows from[13, Theorem 5.1] that the map Q admits a rightward spreading speed c * + and a leftward spreading speed c * − such that the above two statements hold true.Here we have used the property that c * + + c * − > 0, which will be proved in Lemma 3.5.

(4. 8 )
By the comparison arguments, it follows that if 0
with a = 1 being the natural death rate of the adult population and b = 0.01 representing the strength of the density-dependent interspecific competition between individuals.Let g(x, N ) ≡ g(N ) = pN q+N , with p = 1.8 and q = 0.2.The function g(N ) is the Beverton-Holt function, which is monotone with respect to N .The random diffusion functions are fixed at D M = 1 and D I = 0.2.Let the initial function N 0 (x) = cos πx 20 with a compact support on [−10, 10] in the domain [−100, 100].Based on the above parameters and different d I and τ , the spreading of species is obtained, as shown in Figure 2.They demonstrate the effect of d I and τ on the positive steady state of the adult population.More precisely, Figures 2(a) and 2(b) indicate that the positive steady state of the adult population will decrease with τ increasing as d I = 0.5.Observing Figures 2(c) and 2(d) carefully, we can find that the negative correlation between the positive steady state and τ as d I decreases to d I = 0.05.The fundamental reason is the relationship between functions f and d I .In terms of ecology, when |f | > d I , that is, the mortality rate of mature individuals is higher than that of immature individuals.Then the longer it takes for immature individuals to grow into adult individuals (i.e., τ ↗), the higher the positive steady-state of adult groups.When |f | < d I , the correlation is the opposite.

3 .
|f | > d I ; In the second case, adult species are more likely to adapt to the environment and survive in some harsh environments, that is, |f | < d I .This result is an improvement of the numerical simulation in [1].Next we give numerical experiments to show the long-time behaviour of the solution of system (2.1) with time-space periodic parameters in the monotone and non-monotone cases, respectively.Let N 0 (x) = 0.2 cos πx 20 with a compact support on [−10, 10] in the domain [−50, 50], w(t, x) = (1+0.3cos t)(1+0.3sin x), and f (t, x, u) = w(t, x)f (u) = w(t, x)(−au−bu 2 ) with a = 1 and b = 0.01.In the monotone case, we adopt the birth function g as g 1 = pN q+N form, and other parameters remain the same as above.This result is demonstrated in Figure In the non-monotone case, we take g = g 2 = N e r−kN with positive constants r = 2.5 and k = 1, the function g 2 is the Ricker function, which is shown in Figure 4.Both Figures 3 and (a) dI = 0.5 and τ = 0.2.(b) dI = 0.5 and τ = 0.8.(c) dI = 0.05 and τ = 0.2.(d) dI = 0.05 and τ = 0.8.

Figure 2 :
Figure 2: If dI = 0.5, then the positive steady state will decrease when only parameter τ increases from 0.2 to 0.8, (from Figure 2(a) to Figure 2(b)).Nevertheless, if dI = 0.05, then the positive steady state will increase when only parameter τ increases from 0.2 to 0.8, (from Figure 2(c) to Figure 2(d)).

Figure 3 :
Figure 3: The long-time behaviour of the solution of system (2.1) with time-space periodic parameters and the monotone birth function.

4
indicate that the solutions of system (2.1) connect 0 and time-space periodic solutions in the monotone and non-monotone cases, respectively.
(a) The spatial and temporal evolution of adult population.(b) The two-dimensional projection of Figure 4(a) on the x t plane.

Figure 4 :
Figure 4: The long-time behaviour of the solution of system (2.1) with time-space periodic parameters and the non-monotone birth function.
boundary conditions and establish the global stability results.Note that when the bounded domain is one-dimensional with Dirichlet boundary condition, we can also obtain the critical domain size to describe the persistence and extinction of species, which is similar to those in [1, Section 4] and [25, Section 2].The numerical simulations reveal meaningful phenomena when |f | > d I , the positive steady state and time τ are positively correlated, while when |f | < d I , the relationship is converse.