Spreading-vanishing dichotomy in a degenerate logistic model with general logistic nonlinear term
© Dong et al.; licensee Springer 2014
Received: 12 June 2014
Accepted: 4 July 2014
Published: 24 September 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 ).
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 ).
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 .
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 , , , , , .
still represents an elliptic operator acting on () over the ball , whose coefficients are continuous in when .
where is a constant dependent on , and .
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.
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 .
Thus for .
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 the following two lemmas, we can obtain the spreading-vanishing dichotomy.
If, then, and.
Hence , which contradicts our assumption . So we have .
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 . □
Combining Lemmas 3.1 and 3.2, we can easily obtain the spreading-vanishing dichotomy as follows.
- 1.Spreading: and
Vanishing: and .
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).
We denote by the unique solution of the above problem.
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.
To simplify the discussion, we omit from , , , and in the following argument.
This contradicts as . □
Suppose, then there existsdepending onsuch that vanishing occurs when.
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.
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.
- Skellam JG: Random dispersal in theoretical populations. Biometrika 1951, 38: 196-218. 10.1093/biomet/38.1-2.196MathSciNetView ArticleGoogle Scholar
- Shigesada N, Kawasaki K: Biological Invasions: Theory and Practice. Oxford University Press, Oxford; 1997.Google Scholar
- Fisher RA: The wave of advance of advantageous genes. Annu. Eugen. 1937, 7: 335-369.Google Scholar
- Kolmogorov AN, Petrovsky IG, Piskunov NS: Étude de l’équation de la diffusion avec croissance de la quantité de matière et son application a un problème biologique. Bull. Univ. État Mosc. Sér. Int. A 1937, 1: 1-26.Google Scholar
- Aronson DG, Weinberger HF: Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. Partial Differential Equations and Related Topics 1975, 5-49. 10.1007/BFb0070595View ArticleGoogle Scholar
- Taylor CM, Hastings A: Allee effects in biological invasions. Ecol. Lett. 2005, 8: 895-908. 10.1111/j.1461-0248.2005.00787.xView ArticleGoogle Scholar
- Du Y, Lin ZG: Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary. SIAM J. Math. Anal. 2010, 42: 377-405. 10.1137/090771089MathSciNetView ArticleGoogle Scholar
- Du Y, Guo ZM: Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, II. J. Differ. Equ. 2011, 250: 4336-4366. 10.1016/j.jde.2011.02.011MathSciNetView ArticleGoogle Scholar
- Rubinstein LI: The Stefan Problem. Am. Math. Soc., Providence; 1971.Google Scholar
- Cantrell RS, Cosner C: Spatial Ecology via Reaction-Diffusion Equations. Wiley, New York; 2003.Google Scholar
- Chen XF, Friedman A: A free boundary problem arising in a model of wound healing. SIAM J. Math. Anal. 2000, 32: 778-800. 10.1137/S0036141099351693MathSciNetView ArticleGoogle Scholar
- Keller JB:On solutions of . Commun. Pure Appl. Math. 1957, 10: 503-510. 10.1002/cpa.3160100402View ArticleGoogle Scholar
- Osserman R:On the inequality . Pac. J. Math. 1957, 7: 1641-1647. 10.2140/pjm.1957.7.1641MathSciNetView ArticleGoogle Scholar
- Ladyzenskaja OA, Solonnikov VA, Ural’ceva NN: Linear and Quasilinear Equations of Parabolic Type. Am. Math. Soc., Providence; 1968.Google Scholar
- Dong W, Liu L:Uniqueness and existence of positive solutions for degenerate logistic type elliptic equations on . Nonlinear Anal. 2007, 67: 1226-1235. 10.1016/j.na.2006.07.009MathSciNetView ArticleGoogle Scholar
- Du Y, Ma L:Logistic type equations on by a squeezing method involving boundary blow-up solutions. J. Lond. Math. Soc. 2001, 64: 107-124. 10.1017/S0024610701002289MathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd.Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.