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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 201094, 11 pages
Existence of Traveling Wave Solutions for Cholera Model
1School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
2College of Mathematics & Computer Science, Yangtze Normal University, Chongqing 408100, China
Received 28 December 2013; Accepted 25 January 2014; Published 26 March 2014
Academic Editor: Kaifa Wang
Copyright © 2014 Tianran Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
To investigate the spreading speed of cholera, Codeço’s cholera model (2001) is developed by a reaction-diffusion model that incorporates both indirect environment-to-human and direct human-to-human transmissions and the pathogen diffusion. The two transmission incidences are supposed to be saturated with infective density and pathogen density. The basic reproduction number is defined and the formula for minimal wave speed is given. It is proved by shooting method that there exists a traveling wave solution with speed for cholera model if and only if .
Cholera has been a serious threat to human health in the past and at present, which is an acute, diarrheal illness caused by infection of the intestine with the bacterium Vibrio cholera. An estimated 3–5 million cases and over 100,000 deaths occur each year around the world . The cholera bacterium is usually found in water or food sources that have been contaminated by feces from a person infected with cholera. Cholera is most likely to be found and to spread in places with inadequate water treatment, poor sanitation, and inadequate hygiene. Therefore, cholera outbreaks have occurred in developing countries, for example, Iraq (2007-2008), Guinea Bissau (2008), Zimbabwe (2008-2009), Haiti (2010), Democratic Republic of Congo (2011-2012), and Sierra Leone (2012) .
To understand the propagation mechanism of cholera, many mathematical models were proposed, whose earlier one was established by Capasso and Paveri-Fontana  to study the 1973 cholera epidemic in the Mediterranean region as follows: where and denote the concentrations of the pathogen and the infective populations, respectively. In addition, Codeço  investigated the role of the aquatic pathogen in dynamics of cholera through the following susceptible-infective-pathogen model: where is the susceptible individuals. In this model, human is divided into two groups: the susceptible group and the infective group. As pointed out in [4–8], bacterium Vibrio cholera can spread by direct human-to-human and indirect environment-to-human modes. To understand the complex dynamics of cholera, model (2) is extended by [8–15] and so forth.
In all previous models the influences of space distribution of human on the transmission of cholera are omitted. Cholera usually spreads in spatial wave . Cholera bacteria live in rivers and interact with the plankton on the surface of the water . When individuals drink contaminated water and are infected, they will release cholera bacteria through excretion . Capasso et al. [19–23] developed model (1) by incorporating the bacterium diffusion in a bounded area and studied the existence and stability of solutions. To deeply investigate the interaction of transmission modes and bacterium diffusion, Bertuzzo et al. [24, 25] incorporated patchy structure into model (2) and supposed that pathogen in water could diffuse among these patches. Furthermore, Mari et al.  studied the influence of diffusion of both human and pathogen on cholera dynamics through a patchy model.
Infectious case is usually found firstly at some location and then spreads to other areas. Consequently, the most important question for cholera is what the spreading speed of cholera is. However, the above spatial models mainly focus on the stability of solutions not the spreading speed. Traveling wave solution is an important tool used to study the spreading speed of infectious diseases [27–29]. Based on Capasso’s model (1), Zhao and Wang , Xu and Zhao , Jin and Zhao , and Hsu and Yang  studied the influences of pathogen diffusion on the spread speed of cholera.
The studies of traveling wave solutions of Capasso’s model (1) incorporating pathogen diffusion provide insight into the spreading speed of cholera. However, some pieces of information are omitted, such as the interaction of direct human-to-human and indirect environment-to-human transmissions. In this paper, a reaction-diffusion model with pathogen diffusion and both transmission paths is proposed by developing Codeço’s model (2). Based on model (2) and ignoring the disease-related death, a general diffusive cholera model can be formulated as the following reaction-diffusion system: where and denote the concentrations of susceptible and infected individuals, respectively, and is the concentration of the infectious agents. is the total human population, stands for the natural birth and death rate, denotes the contribution of each infected person to the concentration of cholera, and is the net death rate of vibrio cholera. and are the human-to-human and environment-to-human transmission incidences, respectively. Similar to , we assume that and satisfy (A1), , ;(A2), , , , and is strictly monotonously increasing in .It is easy to conclude that , , and and are nonincreasing. Obviously, hypotheses (A1) and (A2) imply that the two transmission paths are saturated. In Tian and Wang , and have the following expressions: Obviously, as a special case, such selections satisfy (A1) and (A2).
Shooting method is very important in proving the existence of traveling wave solutions, which was proposed by Dunbar [34, 35] and was applied to many models (e.g., [36–40]). In this paper, the existence of traveling wave solutions of system (3) will be proved by shooting method and the formula for minimal wave speed will be given.
This paper is organized as follows. In next section, the main theorem and the formula for minimal wave speed will be given. In Section 3, the nonexistence of the traveling wave solutions for is proved by geometric method. Section 4 is devoted to shooting arguments and the construction of Wazewski set. In Section 5, we prove the existence of traveling wave solutions for and then give the existence of traveling wave solution for by limit arguments. The final section is devoted to the simulations.
2. Main Results
For convenience, we introduce dimensionless variables and parameters. By setting model (3) has the form where and .
Denote , which is the basic reproduction number of (6). Then hypotheses (A1) and (A2) imply that system (6) has two nonnegative constant solutions and if and only if , where is the only one positive root of equation and . Biologically, corresponds to disease-free equilibrium and corresponds to endemic equilibrium. To study the spreading wave of cholera, it is assumed that holds in this paper; that is
A traveling wave solution of system (6) with speed is a nonnegative solution of the form
Before giving the main theorem, we introduce the equation for minimal wave speed where
Theorem 1. There exists a constant which is the greatest positive root of (12). When , system (6) has a traveling wave solution satisfying boundary condition (11). When , system (6) has no traveling wave solutions satisfying boundary condition (11).
3. Nonexistence of Traveling Wave Solutions for
From (10), we have
Consequently, if , then
Hence, the traveling profile with boundary condition (11) must satisfy
If , then by system (10). Therefore, we suppose for any ; that is, .
Obviously, system (17) has two equilibria and . A profile solution of (10) which satisfies boundary condition (11) corresponds to the positive solution of system (17) which satisfies where . Therefore, to study the solutions of (10), it is sufficient to study those of system (17) satisfying boundary condition (18).
Firstly, we investigate the dynamics near . Simple calculations show that the characteristic equation of the linearization of system (17) at is where
Because (19) has a negative real root, which is denoted by . Let and be the other two eigenvalues of (19) and suppose that . To investigate the distribution of roots of (19), denote and introduce the following lemma .
Lemma 2. If , (19) has one real root and two nonreal complex conjugate roots.
If , (19) has a multiple root and all its roots are real.
If , (19) has three distinct real roots.
Direct calculations show that , where is defined by (12).
Lemma 3. The real parts of and are positive.
Assume . Then, there exists which is the only positive root of . When , , and are real. When , , and are complex and nonreal.
Assume that . Then, there exist two positive constants which are all positive roots of . and are complex and nonreal if and only if . If , then ; if , then , where .
if and only if or .
Proof. Suppose is the root of (19). Substituting into (19) and comparing real and imaginary parts show that and . Since , then . However, it is impossible that and by the expressions of and . Therefore, the real parts of and are not zero. Furthermore, since it is impossible that and , Routh-Hurwitz theorem implies that it is impossible that the real parts of both and are negative. Consequently, there are two cases: (i) and are complex conjugate roots with positive real parts; (ii) and are real and at least one is positive. However, Descartes’ rule of signs shows that the number of positive roots of (19) is zero or two. Thus, if case (ii) is true, both of and are real and positive. Therefore, (a) is proved.
In this paragraph, we consider the case . Firstly, suppose that . Obviously, and . By the expression of , we have since . Now, assume ; that is, . Then. , , , and . Then, if , Descartes' rule of signs shows that there exists which is the only positive root of , where for and for . Using Lemma 2 completes the proof of (b).
Suppose that in this paragraph and, thus, . Calculations show that and that has two roots and , where and . By letting and using trivial calculations, we get (see Figure 1)
Therefore, if , then . Since is the only minimum-value point of , and then for any and both of and are not real. Lemma 2 shows that . Thus, since and , there exist two positive roots for equation such that . Then, using (25) and Lemma 2 completes the proof of (c) and (d).
Direct calculations show that corresponding eigenvectors of eigenvalue are where . Since and thus
Then, we have the following lemma.
Lemma 4. If , there exist no traveling wave solutions which satisfy boundary condition (11).
Proof. Assume that and . Then, (b) of Lemma 3 implies that and are complex conjugate eigenvalues and there exits locally unstable manifold and locally stable manifold . If a solution of (17) tends to when , then it will be spiral on . By the structures of and , at some time , which shows that there exist no traveling wave solutions departing from .
Suppose that . If , (c) of Lemma 3 shows that and are complex conjugate eigenvalues and similar arguments to that of previous paragraph finish the proof. If , (c) of Lemma 3 shows that and are real; however, . If a solution of (17) tends to when , structures of and indicate that there is an such that . The proof is completed.
4. Shooting Method and Wazewski Set
To prove the existence of traveling wave, shooting method developed by Dunbar  is used. Firstly, we give the shooting arguments.
Consider the differential equation where from to satisfies Lipschitz condition about . Let denote the unique solution of (29) with initial value . It is convenient to give the notations and . To describe the shooting method (or Wazewski theorem), some definitions are necessary.
Definition 5. (a) For , define immediate exit set of as
(b) For , let .
(c) Given , define exit time of by
Then, Wazewski theorem is formulated as follows.
Lemma 6 (see ). Suppose that (1)if and , then .(2)If , and , then there exists an open set about disjoint from .(3)If , is compact and intersects a trajectory of (29) only once.
Then, the mapping is a homeomorphism from to its image on .
A set satisfying conditions (1) and (2) of Lemma 6 is called a Wazewski set. In the following, we first construct the Wazewski set . Fundamental idea to construct a Wazewski set is that the characteristic vectors corresponding eigenvalues with positive real parts should be removed from and that those characteristic vectors corresponding eigenvalues with negative real parts should be included. Therefore, we set where
It is obvious that . Firstly, we give the construction of , which is described in Figure 2.
Lemma 7. The construction of is as follows: where .
Proof. It is enough to analyze the behavior of solution on . We only study and omit the proof of since the analysis of is similar to that of and is simpler. In the process of this proof, we use some notations to simplify the proof. Set
From hypotheses (A1) and (A2), we find that and are monotonously decreasing, is strictly monotonously decreasing for , and is the only positive root of . The set is classified into two cases according to variable . (a) Case . This case is classified as follows. (1) Case . Then and the solution of (17) will enter . (2) Case . Then The solution of (17) will enter . (3) Case . Then The solution of (17) will enter . (4) Case . Then and the solution of (17) will enter . (5) Case and . The solution of (17) will enter . (6) Case . Then , and . Therefore, the solution of (17) will enter . (b) Case . This case is classified as follows. (1) Case . Then and the solution of (17) will enter . (2) Case . Then , . The solution of (17) will enter . (3) Case . Then , , and . The solution of (17) will enter . (4) Case . Then and . The solution of (17) will enter . (5) Case . In this case, is equilibrium and is constant. (6) Case and . Then and . The solution of (17) will enter . (7) Case . Then is equilibrium and is constant.
The proof is completed.
5. Existence of Traveling Wave Solution for
In this section, we prove the existence of traveling wave solution for . Firstly, we study the behaviors of solutions near .
5.1. Behaviors of Solutions Near
Lemma 8. Suppose is a solution of (17) satisfying initial conditions where . Then, for every , we have
Proof. From Lemma 3, we have . To finish the proof, it is sufficient to prove that the set
is positively invariant. It is obvious that
Suppose that . Then, and
The last inequality is given since . Suppose that . If , then
If , we have Consequently, the solution of system (17) departing from cannot intersect . If , then . Since is equilibrium, in summary, is positive invariant.
Since , stable manifold theorem implies that there exists a one-dimensional strong unstable manifold tangent to at such that the point on near can be expressed by Furthermore, there is a two-dimensional unstable manifold tangent to span at such that near can be expressed by
Lemma 9. Suppose that is a solution of (17) such that for small . Then, will leave and enter .
Proof. Obviously, satisfies initial condition (39) by the structure of , and Lemma 8 implies ( means that and , ) for every .
Furthermore, Lemma 8 shows that , implying . Since , it follows . Suppose that for every . Then for large since and are strictly monotonous increasing with respect to and , respectively. Thus, we have that , contradicting for any . Therefore, there exists such that . Without losing generality, let . Obviously, we have . If , thenwhich is a contradiction. Therefore, and . Then, the construction of shows that will leave and enter .
Let be a small circle on centered at . Then, points on can be expressed in terms of local coordinate by where , , and is chosen such that lies on with . Then, stable manifold theorem shows that when . Denote .
Lemma 10. There exists a such that and that for .
Proof. From (51), we have
where , , and since . Therefore, and imply that . Obviously, for any . However,
Then, equality , together with the last of (55), reveals ; that is, . For , the first and second equalities of (55) imply that where since and .
By Lemma 10, is an arc of circle, , and the solution of (17) with initial value being the endpoint will enter since . From Lemma 9, the solution of (17) with initial value being the endpoint will enter .
5.2. Traveling Wave Solution for
Lemma 11. Let be a solution of (17) such that . If for any , then for any , where and .
Proof. Set . Suppose the conclusion is false; that is, . Obviously, and where
In Figure 3, we find , , , , , , and .
Since , thus . If , we have because for and . However, which is a contradiction. Therefore, . If , thencontradicting . If , then since , contradicting . If , then which is a contradiction. In conclusion, . If , then and for any . Hence, for any , which implies that . From this contradiction we find . Because is a constant solution, we get . In summary, and . The proof is completed.
Lemma 12. There exists a point such that the solution of (17) with initial value being will stay in for any .
Proof. It is sufficient to prove . Suppose that . Firstly, we verify Conditions and of Lemma 6. Condition of Lemma 6 is valid since is closed.
Suppose , and . Then, and since . The structure of implies that , , and . By the proof of Lemma 11, we have that for . Therefore, and . Condition of Lemma 6 holds.
Lemma 6 shows that is homeomorphic to . Since and is disconnected, we have that is disconnected, contradicting the connection of . Thus, and the proof is completed.
Lemma 13. Let . Then, there exists a positive solution of (17) such that
Proof. By Lemma 12 there exists a point such that the solution of (17) with initial value being will stay in for any . Furthermore, Lemma 11 shows for any . Stable manifold theorem implies that for any and . Therefore, is a positive solution.
To complete the proof, it is sufficient to show that . By Lemma 11, we know that for any since remains in for all . Because , then the limit of exists; that is, and . Suppose that . The first equation of (17) shows that for large , which implies that there is an such that . This is a contradiction, and thus . From the first equation of (17), we have where is the only positive root of algebra equation At the same time, the third equation of (17) implies and or . It is impossible that due to the boundedness of . In conclusion, the limit exists and is finite. By , must be equilibrium. Since , then .
5.3. Traveling Wave Solution for
Let be a sequence such that for any and . Set and Then, for any .
Proof. It is sufficient to prove that there exists a positive solution of (17) satisfying boundary condition (62).
Firstly, we show that sequences , , , , , and are uniformly bounded and equicontinuous. The idea of Lemma 11 in  is used. Obviously, , , and are uniformly bounded since for any . Because is the solution of (67), , , and are also uniformly bounded. Since where , then is equicontinuous. Similarly, and are also equicontinuous. By differentiating the equations of (67) and using the previous bounds, we can get that , , and are uniformly bounded, and hence , , and are equicontinuous.
The previous paragraph and Arzelà-Ascoli theorem imply that there exist subsequences, again denoted by , , and and functions , , and such that uniformly on compact subsets of , thus pointwise on . Same arguments imply that , , and are also uniformly convergent on compact subsets of and pointwise convergent on . Consequently, we get Since is the solution of (67), then is the solution of (17) for and , where is the closer of . Because system (67) is autonomous and satisfies boundary condition (62), we can assume that for all ; thus, . Then, similar to the proof of Lemma 13, we have that the solution satisfies boundary condition (62).
In this section, we present some simulations to confirm the theoretical results. Set and assign numerical values to parameters as follows:
Obviously, such selection for and satisfies (A1) and (A2). Then, the traveling wave solution is described in Figure 4.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors are supported by the Fundamental Research Funds for the Central Universities (Grants nos. XDJK2012C042 and SWU113048) and NSFC (Grant no. 11201380).
- Centers for Disease Control and Prevention, USA, http://www.cdc.gov/cholera/general/.
- World Health Organization, http://www.who.int/csr/don/archive/disease/cholera/en/.
- V. Capasso and S. L. Paveri-Fontana, “Mathematical model for the 1973 cholera epidemic in the European Mediterranean region,” Revue d'Epidemiologie et de Sante Publique, vol. 27, no. 2, pp. 121–132, 1979.
- C. T. Codeço, “Endemic and epidemic dynamics of cholera: the role of the aquatic reservoir,” BMC Infectious Diseases, vol. 1, no. 1, article 1, 2001.
- J. R. Andrews and S. Basu, “Transmission dynamics and control of cholera in Haiti: an epidemic model,” The Lancet, vol. 377, no. 9773, pp. 1248–1255, 2011.
- K. T. Goh, S. H. Teo, S. Lam, and M. K. Ling, “Person-to-person transmission of cholera in a psychiatric hospital,” Journal of Infection, vol. 20, no. 3, pp. 193–200, 1990.
- A. A. Weil, A. I. Khan, F. Chowdhury et al., “Clinical outcomes in household contacts of patients with cholera in Bangladesh,” Clinical Infectious Diseases, vol. 49, no. 10, pp. 1473–1479, 2009.
- J. H. Tien and D. J. D. Earn, “Multiple transmission pathways and disease dynamics in a waterborne pathogen model,” Bulletin of Mathematical Biology, vol. 72, no. 6, pp. 1506–1533, 2010.
- Z. Mukandavire, S. Liao, J. Wang, H. Gaff, D. L. Smith, and J. G. Morris Jr., “Estimating the reproductive numbers for the 2008-2009 cholera outbreaks in Zimbabwe,” Proceedings of the National Academy of Sciences of the United States of America, vol. 108, no. 21, pp. 8767–8772, 2011.
- J. P. Tian and J. Wang, “Global stability for cholera epidemic models,” Mathematical Biosciences, vol. 232, no. 1, pp. 31–41, 2011.
- D. M. Hartley, J. G. Morris Jr., and D. L. Smith, “Hyperinfectivity: a critical element in the ability of V. cholerae to cause epidemics?” PLoS Medicine, vol. 3, no. 1, pp. 63–69, 2006.
- Z. Shuai and P. van den Driessche, “Global dynamics of cholera models with differential infectivity,” Mathematical Biosciences, vol. 234, no. 2, pp. 118–126, 2011.
- Z. Shuai, J. H. Tien, and P. van den Driessche, “Cholera models with hyperinfectivity and temporary immunity,” Bulletin of Mathematical Biology, vol. 74, no. 10, pp. 2423–2445, 2012.
- M. A. Jensen, S. M. Faruque, J. J. Mekalanos, and B. R. Levin, “Modeling the role of bacteriophage in the control of cholera outbreaks,” Proceedings of the National Academy of Sciences of the United States of America, vol. 103, no. 12, pp. 4652–4657, 2006.
- R. P. Sanches, C. P. Ferreira, and R. A. Kraenkel, “The role of immunity and seasonality in cholera epidemics,” Bulletin of Mathematical Biology, vol. 73, no. 12, pp. 2916–2931, 2011.
- A. Mutreja, D. W. Kim, N. R. Thomson et al., “Evidence for several waves of global transmission in the seventh cholera pandemic,” Nature, vol. 477, no. 7365, pp. 462–465, 2011.
- R. R. Colwell, “Global climate and infectious disease: the cholera paradigm,” Science, vol. 274, no. 5295, pp. 2025–2031, 1996.
- S. M. Faruque, M. J. Islam, Q. S. Ahmad et al., “Self-limiting nature of seasonal cholera epidemics: role of host-mediated amplification of phage,” Proceedings of the National Academy of Sciences of the United States of America, vol. 102, no. 17, pp. 6119–6124, 2005.
- S. Aniţa and V. Capasso, “Stabilization of a reaction-diffusion system modelling a class of spatially structured epidemic systems via feedback control,” Nonlinear Analysis. Real World Applications, vol. 13, no. 2, pp. 725–735, 2012.
- V. Capasso, “Asymptotic stability for an integro-differential reaction-diffusion system,” Journal of Mathematical Analysis and Applications, vol. 103, no. 2, pp. 575–588, 1984.
- V. Capasso and K. Kunisch, “A reaction-diffusion system arising in modelling man-environment diseases,” Quarterly of Applied Mathematics, vol. 46, no. 3, pp. 431–450, 1988.
- V. Capasso and L. Maddalena, “Convergence to equilibrium states for a reaction-diffusion system modelling the spatial spread of a class of bacterial and viral diseases,” Journal of Mathematical Biology, vol. 13, no. 2, pp. 173–184, 1981-1982.
- V. Capasso and R. E. Wilson, “Analysis of a reaction-diffusion system modeling man-environment-man epidemics,” SIAM Journal on Applied Mathematics, vol. 57, no. 2, pp. 327–346, 1997.
- E. Bertuzzo, S. Azaele, A. Maritan, M. Gatto, I. Rodriguez-Iturbe, and A. Rinaldo, “On the space-time evolution of a cholera epidemic,” Water Resources Research, vol. 44, no. 1, Article ID W01424, 2008.
- E. Bertuzzo, R. Casagrandi, M. Gatto, I. Rodriguez-Iturbe, and A. Rinaldo, “On spatially explicit models of cholera epidemics,” Journal of the Royal Society Interface, vol. 7, no. 43, pp. 321–333, 2010.
- L. Mari, E. Bertuzzo, L. Righetto et al., “Modelling cholera epidemics: the role of waterways, human mobility and sanitation,” Journal of the Royal Society Interface, vol. 9, no. 67, pp. 376–388, 2012.
- O. Diekmann, “Thresholds and travelling waves for the geographical spread of infection,” Journal of Mathematical Biology, vol. 6, no. 2, pp. 109–130, 1978.
- M. Lewis, J. Rencławowicz, and P. van den Driessche, “Traveling waves and spread rates for a West Nile virus model,” Bulletin of Mathematical Biology, vol. 68, no. 1, pp. 3–23, 2006.
- J. Radcliffe and L. Rass, “The asymptotic speed of propagation of the deterministic nonreducible -type epidemic,” Journal of Mathematical Biology, vol. 23, no. 3, pp. 341–359, 1986.
- X.-Q. Zhao and W. Wang, “Fisher waves in an epidemic model,” Discrete and Continuous Dynamical Systems B, vol. 4, no. 4, pp. 1117–1128, 2004.
- D. Xu and X.-Q. Zhao, “Bistable waves in an epidemic model,” Journal of Dynamics and Differential Equations, vol. 16, no. 3, pp. 679–707, 2004.
- Y. Jin and X.-Q. Zhao, “Bistable waves for a class of cooperative reaction-diffusion systems,” Journal of Biological Dynamics, vol. 2, no. 2, pp. 196–207, 2008.
- C.-H. Hsu and T.-S. Yang, “Existence, uniqueness, monotonicity and asymptotic behaviour of travelling waves for epidemic models,” Nonlinearity, vol. 26, no. 1, pp. 121–139, 2013.
- S. R. Dunbar, “Travelling wave solutions of diffusive Lotka-Volterra equations,” Journal of Mathematical Biology, vol. 17, no. 1, pp. 11–32, 1983.
- S. R. Dunbar, “Traveling wave solutions of diffusive Lotka-Volterra equations: a heteroclinic connection in R4,” Transactions of the American Mathematical Society, vol. 286, no. 2, pp. 557–594, 1984.
- W.-T. Li and S.-L. Wu, “Traveling waves in a diffusive predator-prey model with Holling type-III functional response,” Chaos, Solitons & Fractals, vol. 37, no. 2, pp. 476–486, 2008.
- C.-H. Hsu, C.-R. Yang, T.-H. Yang, and T.-S. Yang, “Existence of traveling wave solutions for diffusive predator-prey type systems,” Journal of Differential Equations, vol. 252, no. 4, pp. 3040–3075, 2012.
- W. Huang, “Traveling wave solutions for a class of predator-prey systems,” Journal of Dynamics and Differential Equations, vol. 24, no. 3, pp. 633–644, 2012.
- J. Huang, G. Lu, and S. Ruan, “Existence of traveling wave solutions in a diffusive predator-prey model,” Journal of Mathematical Biology, vol. 46, no. 2, pp. 132–152, 2003.
- X. Lin, P. Weng, and C. Wu, “Traveling wave solutions for a predator-prey system with sigmoidal response function,” Journal of Dynamics and Differential Equations, vol. 23, no. 4, pp. 903–921, 2011.
- R. S. Irving, Integers, Polynomials, and Rings, Springer, New York, NY, USA, 2004.
- Z. Ma and Y. Zhou, Qualitative and Stable Method of Ordinary Differential Equations, Science Press, Beijing, China, 2005.