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Spreading-vanishing dichotomy in a degenerate logistic model with general logistic nonlinear term
Boundary Value Problems volume 2014, Article number: 180 (2014)
In this paper, we study the degenerate logistic equation with a free boundary and general logistic term in higher space dimensions and heterogeneous environment, which is used to describe the spreading of a new or invasive species. We first prove the existence and uniqueness of the local solution for the free boundary problem by the contraction mapping theorem, then we show that the solution can be expanded to all time using suitable estimates. Finally, we prove the spreading-vanishing dichotomy.
It is an important problem to study the spreading of the invasive species in invasion ecology, which is an interesting branch of ecology. Using differential equations to study ecology becomes a main approach in ecological research. Most of the ecological phenomena such as species extinction can be explained by the nature of the differential equations. In the research of the spreading of the muskrat in Europe, Skellam observed the well-known phenomenon that many animal species spread to a new environment in a linear speed, which means the spreading radius eventually shows a linear growth speed against times . Firstly, he calculated the square root of the area of the muskrat range from a map, which gives the spreading radius. Then he plotted it against times and observed the data points lay on a straight line. Several mathematical models have been proposed to discuss this phenomenon (see ).
The most successful mathematical model to describe the problem is the following logistic equation over the entire space :
where stands for the population density of a spreading species, is the diffusion rate, is the intrinsic growth rate, means the habitat carrying capacity. Fisher  and Kolomogorov et al. made a pioneering contribution on this problem. They proved the problem admits traveling wave solutions of the problem (1) for space dimension : for any , there is a solution satisfying
or there is no such solution if . The constant is regarded as the minimal speed of the traveling waves. Fisher claimed that the constant is the spreading speed for the advantageous gene and proved it by a probabilistic argument. Then Aronson and Weinberger gave a clearer description and a rigorous proof for this phenomenon (see ).
Although the approach predicts the successful spreading and the establishment of a new species with a nontrivial initial population , it has the obvious shortcoming that it is regardless of its initial population and the initial area, which is in sharp contrast with the real-world observations. It is a well-known conclusion that the large time behavior of a species determined by its initial size. In the real-world environment, an animal species either profits from the same species, or it will be hurt by the competition with the same species. This phenomenon is called ‘Allee effect’, namely there exists a critical population density such that the species can establish itself when the density is greater than the critical value, or it will die out on the other hand . To include the ‘Allee effect’, we usually replace the logistic term in (1) by a bistable function as follows:
In 2010, Du used a free boundary problem to describe the spreading of species in , . First, Du studied the problem in one space dimension and with a homogeneous environment in  and then extended the conclusions to higher space dimensions and a heterogeneous environment in . Considering the following free boundary problem with logistic term in the same way as the problem (1), Du proved both spreading and vanishing can happen depending on the initial size:
where , , (), , and is the moving boundary to be determined, and the initial function satisfies
Problem (2) describes spreading of a new species over an -dimensional habitat with an initial population density , which occupies an initial region . (Here stands for the ball with the center at 0 and radius .) The free boundary stands for the spreading front, which is the boundary of the ball . The radius of the free boundary increases with a speed that is proportional to the population gradient at the front: . In the same way as (1), the coefficient function means an intrinsic growth, represents an intra-specific competition, and is the diffusion rate.
The free boundary is governed by the equation , which is a special case of the well-known Stefan condition. The condition has been applied in a number of problems. For example, it was used to describe the melting of ice in contact with water , in the modeling of oxygen in the muscle , and in wound healing .
Du has proved that the problem (2) admits a unique solution for all the with and . Moreover, compared with the traditional logistic equation, the solution of the free boundary problem (2) is typical of the spreading-vanishing dichotomy. All this means that as the species either successfully spreads to the entire new environment and stabilizes at a positive equilibrium (called spreading), in the case that and , or it fails to establish itself and dies out in the long run (called vanishing), in the sense that and . The criteria for spreading or vanishing are as follows: If the radius of the initial region is greater than a critical size , namely , then the spreading always occurs for all the initial function satisfying (3). On the other hand, if , whether spreading or vanishing happens is determined by the initial population and the coefficient in the Stefan condition.
Compared with the free boundary problem (2) and the problem (1), (2) is more similar to the spreading process in real world. At first, compared with the persistent spreading in the model (1), both spreading and vanishing can occur in the model (2) depending on the initial size. Next, for any finite , the solution of the problem (2) is supported on a finite domain of , which expands with the increase of . However, in the problem (1), the solution is always positive for all the as .
The logistic term of the form has been thoroughly discussed by Du in , . In ecology, this logistic term is too simple to describe the phenomenon in the real world. Thus, we will study a more complex logistic term as follows:
Here, the main condition is the same as the problem (2), where , , (), , is a moving boundary to be determined, , and are given positive constants, , , and there exist positive constants such that
The initial function satisfies
Moreover, the logistic nonlinear term satisfy the conditions (A1) and (A2) listed below:
(A1) and is increasing on ;
(A2), where .
Keller  and Osserman  proposed these conditions in 1957. These conditions have been used widely to study those functions which behave like (). We can easily obtain , from condition (A2). Clearly, is a special case.
In Section 2, we first prove the existence and uniqueness of the local solution for the free boundary problem (4) (Theorem 2.1) by the contraction mapping theorem, then we show that the solution can be expanded to all using suitable estimates (Theorem 2.3). Finally, we prove the spreading-vanishing dichotomy in Section 3.
2 Existence and uniqueness for the free boundary problem
where, andonly depend on, and.
where , , , , , .
We denote and for ,
It is easily seen that is a complete metric space with the following metric:
For , due to , we have
Next, we use the contraction mapping theorem to prove the existence and uniqueness of the local solution. Firstly, for any given , we have
Thus, for , we have and for such ,
So, although is singular at ,
still represents an elliptic operator acting on () over the ball , whose coefficients are continuous in when .
Applying theory and the Sobolev imbedding theorem , we find that the following initial boundary value problem:
has a unique solution for any and
where is a constant dependent on , and .
Now, we define ,
Clearly is a solution of (8) if and only if it is the fixed point of ℱ.
Thus, if we let , ℱ maps into itself.
Next, we will prove that if is sufficiently small, ℱ is a contraction mapping on . In fact, if we let and denote , we obtain
Setting , then satisfies
Using the estimates and the Sobolev imbedding theorem, we have
where depends on , and , , , . Taking the difference of the equations for , results in
where depends on and . Therefore for
This means that ℱ is a contraction mapping on . By the contraction mapping theorem, we find that ℱ has a unique fixed point in . Moreover, it follows that we have the Schauder estimates and . Moreover, we have (11) and (13). This shows that is a unique local classical solution of the problem (8). □
Next, we will use some suitable estimates to show that the solution can be extended to all .
Letbe the solution to the problem (2) defined onfor some. Then there exist constantsandindependent ofsuch that
By the strong maximum principle, we have
Thus for .
Due to (5), using the comparison principle, we have for , , where is the solution of following problem:
By the condition (A2), we easily obtain , . Thus, there exists an such that . Clearly, the is the supremum of the problem (16). Therefore, we have
Next, using the approach in , it is easy to prove that for , where is independent of . Then the proof is complete. □
The proof is the same as Theorem 4.3 in . So we omit the details.
3 Spreading-vanishing dichotomy
By Lemma 2.2, we see that is monotonic and therefore there exists such that . Let be the principal eigenvalue of the problem
It is well known that is a strictly decreasing continuous function and
Thus, for fixed and , there is a unique such that
By the following two lemmas, we can obtain the spreading-vanishing dichotomy.
If, then, and.
We first prove . Arguing by contradiction, we suppose and there is a such that for all the . Therefore, for all , we have
Moreover, for any sufficiently small , there exists a such that
Consider the following problem:
where is the unique positive solution of the following problem:
Using the comparison principle
which implies that
On the other hand, consider the problem
Clearly, the problem (24) also has a unique positive solution
where is a unique positive solution of the following problem:
In the same way, the comparison principle implies that
Using a compactness and uniqueness argument, we can easily obtain
By the argument of Lemma 2.2 in , we obtain
which implies that
Hence , which contradicts our assumption . So we have .
Next, we will prove that as . Let be the unique positive solution of the following problem:
The comparison principle implies for and . Due to , we have and it follows from a well-known conclusion about logistic equation that uniformly for as (see ). Therefore, we get . □
whereis the unique positive solution of the following problem:
and the boundary blow-up problem
Choose an increasing sequence such that as , and for all . Then both and converge to as . For each , we can find a such that for . Note that the following problem:
has a unique positive solution , and
Using the comparison principle, we have
Let , we have
Similarly, by the proof of Theorem 4.1 in , we obtain
Let , we have
Combining Lemmas 3.1 and 3.2, we can easily obtain the spreading-vanishing dichotomy as follows.
Letbe the solution of the free boundary problem (2). Then the following alternative holds:
Vanishing: and .
Next, we will discuss when the two alternatives occur exactly. We divide the argument into two cases:
In case (a), we can easily obtain since for all . Therefore, the following conclusion follows from Lemma 3.1.
In the same way as in the discussion in , we need a comparison principle which can be used to estimate both and the free boundary to study case (b).
Suppose, , with, and
If we have
then the solutionof the free boundary problem (4) satisfies
For small , in the problem (4), let replace , replace , and replace , where satisfying
We denote by the unique solution of the above problem.
We claim that for all . Clearly, it is true for small . If our claim is wrong, then we will find a first such that for and , which implies
Now, we compare and over the region . Using the strong maximum principle in , we have . Thus with . It follows that . Then we obtain . Due to , it follows that , which contradicts (37). This shows our claim is correct. Then applying the usual comparison principle for , we obtain .
Now, we consider case (b). As in , we first examine the case that is large, then we investigate the case is small. Finally, we use Lemma 3.5 to show that there exists a critical such that spreading occurs when and vanishing happens when .
Suppose, then there existsdepending onsuch that spreading occurs when.
We prove it by contradiction. Suppose that there is an increasing sequence satisfying as such that the unique solution of the problem (4) with satisfies for all the . Thus, by Lemma 3.1, we have and hence
where is the unique solution of following problem:
Due to the fact that , we have
To simplify the discussion, we omit from , , , and in the following argument.
One calculates directly
Integrating from to implies
For and , due to , we have
Using Lemma 3.5, and are increasing in . Thus
Thus, from (40) we deduce
This contradicts as . □
Suppose, then there existsdepending onsuch that vanishing occurs when.
At first, we will construct a suitable upper solution of the problem (2), then we use Lemma 3.5 to obtain the conclusion. For and , define
where , , are positive constants to be chosen later and is the first eigenfunction of the following problem:
with and . Due to , we have
Since we have the fact that and
Let , then . Direct calculation gives
Due to condition (A1), we easily obtain . Since , we can choose sufficiently small such that
Let , we have
Now, we choose sufficiently large such that
Direct calculation implies
Thus if we let
then for any , we have
Hence, from Lemma 3.5, we have and for , . This implies . □
In the same way as the proof of Theorem 2.1 in , we can prove the following theorem.
If, then there exists adepending onsuch that spreading occurs when, and vanishing happens when.
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The authors thank the referees much for their helpful suggestions. This work was supported by the China Postdoctoral Science Foundation and the Heilongjiang Province Postdoctoral Science Foundation.
The authors declare that they have no competing interests.
XD provided the main ideas and lead to write the paper. YL and XJ proved the local existence part. LM and KS proved the rest. HQ checked all the arguments. All authors read and approved the final manuscript.
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Dong, X., Li, Y., Jiang, X. et al. Spreading-vanishing dichotomy in a degenerate logistic model with general logistic nonlinear term. Bound Value Probl 2014, 180 (2014). https://doi.org/10.1186/s13661-014-0180-9
- logistic equation
- free boundary problem
- spreading-vanishing dichotomy
- invasion ecology