Asymptotic behavior of solutions of a free boundary model with seasonal succession and impulsive harvesting
-
Yanglei Li
Abstract
In this article, we consider the influence of seasonal succession and impulsive harvesting on the dynamical behavior of solutions to a free boundary model. First, the generalized principal eigenvalue is defined and its properties are studied. Next, some sufficient conditions for spreading and for vanishing are given. Then, by introducing a one-parameter family of initial data
1 introduction
The purpose of this article is to consider the combined effects of seasonal succession and impulsive harvesting on the dynamical behavior of solutions to the following problem:
Problem (P) can be used to model the spreading of an invasion species, whose growth and survival are affected by seasonal succession and impulsive harvesting, and its density is stood for
the harvesting function
In the special case that
This problem was innovated and studied in [7]. It was shown in [7] that either spreading happens (i.e.,
They also found that as
When
Problem (1.3) was introduced and discussed by Peng and Zhao [25]. They examined the effect of the seasonal succession on the dynamical behavior of solutions and found that when
When
The above problem was ushered and investigated by Meng et al. [21]. They considered the effect of impulsive harvesting on the dynamical behavior of solutions and showed that if
The main purpose of this article is to study the effects of seasonal succession and impulsive harvesting on the dynamical behavior of solutions to problem (P). When
Moreover, we check that
Our first main result is devoted to the asymptotic behavior of solutions to problem (P), which is stated as follows.
Theorem 1.1
Assume that
when
when
From the above theorem and its proof, we see that there is a critical value
When spreading happens, it is interesting to study the asymptotic spreading speed. It is seen from Section 4 that the following problem:
admits a unique positive solution
Theorem 1.2
Assume that
The proof of this theorem is given in Section 4, which is divided into two parts: the first part covers the existence and uniqueness of positive solutions to problem (1.5); in the second part we construct some suitable super- and sub-solutions of problem (P) to prove (1.6).
The rest of the article is organized as follows. Section 2 covers some basic results which are useful for this research, and may have other applications. In Section 3, we study the long-time behavior of solutions to problem (P). Section 3.1 is filled with the vanishing phenomenon. Section 3.2 is devoted to the spreading phenomenon. The proof of Theorem 1.1 is given in Section 3.3. In Section 4, we show that problem (1.5) admits a unique positive solution
2 Some basic results
In this section, we give some basic results which will be frequently used later in the article.
2.1 Comparison principle
In this subsection, we establish the comparison principle, which will be used in the rest of this article. Let us start with the following result.
Lemma 2.1
Assume that(H) holds,
If
Lemma 2.2
Assume that(H) holds,
where
By a similar argument to that of [7, Lemma 5.7], we can prove Lemma 2.1, and a minor modification of this proof implies Lemma 2.2.
Remark 2.3
The function
2.2 Linear eigenvalue problem
For any fixed
Let us define the generalized principal eigenvalue
where
We obtain the following result on the properties of the generalized principal eigenvalue of problem (2.1).
Lemma 2.4
For any fixed
If
If
(2.4)such that
Proof
First, let us show that
where
we can check that
Next we prove that
Multiplying the first two inequalities in problem (2.2) with
It follows from the divergence formula, the boundary condition, and the periodic condition that
This, together with (2.8) and the fact that
(i) If
(ii) If
3 Classification of the long-time behavior and sharp thresholds
The purpose of this section is to study the long-time behavior of solutions. The first subsection covers the vanishing phenomenon. The second subsection deals with the spreading phenomenon. The proof of Theorem 1.1 is given in the last subsection.
3.1 Vanishing phenomenon
This subsection starts with the following result.
Lemma 3.1
Assume that
Proof
“(i)
where
where
Recalling that
for some
“(ii)
“(ii)
with
“(iii)
Thanks to Lemma 2.4, we know that problem (2.1) with
where
We can also choose
The comparison principle can be used to show that
which is in contradiction to assertion (iii). The proof of this lemma is complete.□
Next, a sufficient condition for vanishing is given as follows.
Lemma 3.2
Assume that
Proof
For any fixed
Choose
then we can find
Define
A direct calculation shows that
Moreover, we can compute that for
As
then
it thus follows from Lemma 2.1 that
This, together with Lemma 3.1, implies that vanishing happens. The proof is complete.□
3.2 Spreading phenomenon
The purpose of this subsection is to study the spreading phenomenon. Let us start with the following result.
Lemma 3.3
Assume that
where
Proof
First, we show the existence and uniqueness of the positive solution
As
where
It is direct to check that problem (3.6) admits a positive solution if and only if
Since
Thus, the existence of positive solution to problem (3.6) is proved. Combining with the monotonicity of
Next, since
where
Since
admits a unique positive solution
Moreover, as
Later, it follows from the proof of [13, Lemma 2.1] that the unique solution
satisfies
This, together with (3.12) and the periodicity of
Later, we show the following sufficient condition for spreading.
Lemma 3.4
Assume that
Proof
Noting that
3.3 Proof of Theorem 1.1
By Lemma 3.1, we see that either
Set
When
This, together with Lemma 3.4, produces that spreading happens for
4 Asymptotic spreading speeds
Throughout this section we assume that
4.1
T
-periodic traveling semi-waves
This subsection is devoted to the existence and uniqueness of
For any nonnegative
We obtain the following result.
Theorem 4.1
Assume that
Moreover, the unique positive solution
Proof
The proof is similar to that in [6] (see also [30]), we give the details for the sake of completeness. Our argument is divided into five steps.
Step 1. Existence of a positive and increasing solution when nonnegative
Let us consider the following linear eigenvalue problem:
It follows from [6,21,23] and Lemma 2.4 that problem (4.2) admits a generalized principal eigenvalue
where
We can check that
In fact,
where
Moreover, by a similar argument as that in Lemma 2.4, we can obtain that
When
Then we can check that
Step 2. Non-existence when
Assume that
where
Then
Suppose by way of contradiction that problem (4.1) admits a positive solution
Clearly
Set
where the monotonicity of
which yields that
Step 3. Uniqueness when
Let us fix
We claim that
where
This is in contradiction to the definition of
Step 4. Monotonicity of positive solution
For any given two functions
We claim that
which implies the monotonicity of
Step 5. Continuity of positive solution
For any nonnegative
In fact, thanks to Steps 1–3, we see that when
where
Thanks to Steps 1 and 3, we can check that problem (4.1) with
which is in contradiction to the fact that
At last, combining with the regularity of parabolic equations, we see that (4.7) holds, so the continuity of
The proof of this theorem is complete now.□
Thanks to the above theorem, we know that for any fixed
where
Lemma 4.2
Under the assumptions of
Theorem 4.1, for any fixed
where
Proof
Thanks to Theorem 4.1, we see that for any fixed nonnegative function
It is direct to compute that
By using the Schauder fixed point theorem we can find some
Moreover, one can check that
Recalling that
The rest of this proof is devoted to the uniqueness of
which is a contradiction. Similarly,
Without loss of generality, we may assume that
is well-defined and
It is direct to check that
Set
Next let us consider the dependence of
Proposition 4.3
Let
when
(4.11)
Proof
(i) For any given
Thanks to Step 4 in the proof of Theorem 4.1, we see that
where
On the one hand, up to a subsequence we see that
it thus follows that
In what follows let us show that
which is in contradiction to
(ii) It follows from the definition of
It is seen from (4.13) that either
(iii) Since
In what follows, let us prove
The comparison principle can be used to infer that
Let us define
then the
We claim that
It is direct to check that
It follows from [16,19,37] that
The proof of this proposition is complete.□
4.2 Asymptotic spreading speed
This section deals with the estimate for the asymptotic spreading speed of two free boundaries and gives the proof of Theorem 1.2.
Proof of Theorem 1.2
We only prove (1.6) for
Let
Recalling that
Let us define
and check that
and for
Clearly,
and for
where
where we have used the monotonicity of
Combining this with the definition of
Letting
On the other hand, we define
and check that
This, together with the fact that
and
When
Noting that
and for
where the fact that
where the monotonicity of
Thanks to this and the definition of
Setting
This, together with (4.15), yields that
Hence the desired result follows.□
5 Discussion
In the present article, a free boundary model with seasonal succession and impulsive harvesting is studied, and this model can be used to describe the biological invasion of the species in a new environment, whose growth and survival is affected by seasonal succession and impulsive harvesting. The main contribution of this article is the characterization of the dynamical behavior of solutions, which is stated in Theorem 1.1. The second contribution is that we use the
The assumption
In the current work, we have assumed that the species live in a one-dimensional homogeneous space. However, the habitat of a biological population, in general, can be rather complicated. For instance, natural river systems are often in a spatial network structure such as dendritic trees. The network topology can greatly influence the species persistence and extinction. Therefore, as in [15,26,27], it would be interesting to consider a more general river habitat (bounded or unbounded) consisting of more than one branch. We plan to study these problems with seasonal succession and impulsive harvesting in future work.
Acknowledgments
The authors would like to thank the anonymous referees for their careful reading and helpful comments, which led to an improvement of our original manuscript.
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Funding information: This work was supported by Shandong Provincial Natural Science Foundation of China (No. ZR2023YQ002), and the Support Plan for Outstanding Youth Innovation Team in Shandong Higher Education Institutions (No. 2021KJ037).
-
Author contributions: Both authors contributed equally and significantly to this manuscript, reviewed all the results and approved the final version of the manuscript.
-
Conflict of interest: The authors state no conflict of interest.
-
Data availability statement: No data were used in this study.
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- Boundedness of solutions to quasilinear elliptic systems
- Existence of positive solutions for critical p-Laplacian equation with critical Neumann boundary condition
- Non-local diffusion and pulse intervention in a faecal-oral model with moving infected fronts
- Nonsmooth analysis of doubly nonlinear second-order evolution equations with nonconvex energy functionals
- Qualitative properties of solutions to the viscoelastic beam equation with damping and logarithmic nonlinear source terms
- Shape of extremal functions for weighted Sobolev-type inequalities
- One-dimensional boundary blow up problem with a nonlocal term
- Doubling measure and regularity to K-quasiminimizers of double-phase energy
- General solutions of weakly delayed discrete systems in 3D
- Global well-posedness of a nonlinear Boussinesq-fluid-structure interaction system with large initial data
- Optimal large time behavior of the 3D rate type viscoelastic fluids
- Local well-posedness for the two-component Benjamin-Ono equation
- Self-similar blow-up solutions of the four-dimensional Schrödinger-Wave system
- Existence and stability of traveling waves in a Keller-Segel system with nonlinear stimulation
- Existence and multiplicity of solutions for a class of superlinear p-Laplacian equations
- On the global large regular solutions of the 1D degenerate compressible Navier-Stokes equations
- Normal forms of piecewise-smooth monodromic systems
- Fractional Dirichlet problems with singular and non-locally convective reaction
- Sharp forced waves of degenerate diffusion equations in shifting environments
- Global boundedness and stability of a quasilinear two-species chemotaxis-competition model with nonlinear sensitivities
- Non-existence, radial symmetry, monotonicity, and Liouville theorem of master equations with fractional p-Laplacian
- Global existence, asymptotic behavior, and finite time blow up of solutions for a class of generalized thermoelastic system with p-Laplacian
- Formation of singularities for a linearly degenerate hyperbolic system arising in magnetohydrodynamics
- Linear stability and bifurcation analysis for a free boundary problem arising in a double-layered tumor model
- Hopf's lemma, asymptotic radial symmetry, and monotonicity of solutions to the logarithmic Laplacian parabolic system
- Generalized quasi-linear fractional Wentzell problems
- Existence, symmetry, and regularity of ground states of a nonlinear Choquard equation in the hyperbolic space
- Normalized solutions for NLS equations with general nonlinearity on compact metric graphs
- Boundedness and global stability in a predator-prey chemotaxis system with indirect pursuit-evasion interaction and nonlocal kinetics
- Review Article
- Existence and stability of contact discontinuities to piston problems
