Global dynamics of population-toxicant models with nonlocal dispersals
-
Li Ma
Abstract
In the article, we first investigate a population-toxicant model with nonlocal dispersal. Compared to the local dispersal (random dispersal), nonlocal dispersal brings some difficulties. One of the difficulties is that principal eigenvalue might not exist. Another difficulty is that the solution orbits lose their compactness. Motivated by previous studies, we overcome these difficulties and establish the global dynamics by spectral analysis, the methods of lower and upper solutions, and the theory of abstract monotone dynamical system. Then, we generalize these results to more general case that the dispersal strategies are mixture of local and nonlocal dispersals. Finally, we point out that we have successfully expanded the upper bound of the critical indicator given in the study by Zhou and Huang without advection terms.
1 Introduction
As is well known, water pollution issues are one of the increasingly hot topics of global concern. In this case, toxic substances are discharged into rivers, streams, lakes, and other body of water, such as pesticides, plastics, pathogens, and chemicals. Toxins in environments bring adverse effects on various levels of ecosystems, including Biodiversity, population growth, community structure, etc. From an environmental and protection perspective, risk assessments of exposed populations are crucial in polluted ecosystems.
In order to better study pollution issues, many mathematicians and ecologists propose appropriate mathematical models on this topic. In these classical models, some include matrix population models (e.g., [11,19,34–36]) and also the ordinary differential equation model (e.g., [13,15–17,22,23]), and their purpose is to study how environmental toxicants affect populations in polluted aquatic ecosystems. In recent decades, researchers have found that many organisms exhibit diffusion phenomena, this makes diffusion mechanisms an ecology’s most important topic. As early as 1951, Skellam [33] first introduced random diffusion to represent dispersal strategies, and there exists a wealth of studies on the subject [5,32]. A number of researchers have realized that how ecological communities are shaped by spatial interactions is a challenge both theoretically and empirically, and understanding the role of space has been identified as an important factor [10,29]. Researching the effects of environmental toxicants on the dynamics of river populations, Zhou and Huang [40] proposed a toxic model with advection term in one dimension domain:
where
Recently, Deng et al. [9] proposed the following model with negative toxicant-taxis to consider the case that moving from areas with high levels of toxicants to areas with low levels of toxicants may improve survival, growth, and reproduction chances
where
While random dispersal is widely used in biological models, it is clearly oversimplified when it comes to describing the movements of many organisms. However, it is well known in ecology that long range dispersals may occur [4,6,7], typical examples include birds flying, seeds spreading, and pollen spreading. In response to this motivation, mathematicians have introduced a novel diffusion mode-nonlocal dispersal, which differs from traditional random diffusion. A widely adopted formulation that incorporates such long-range dispersal is represented by the following nonlocal diffusion operator [3,12,14,24]:
where the kernels
It is natural for us to consider what is the nature of the interaction between a population and a toxicant in contaminated environments when we take the nonlocal dispersal factor into account? In what ways does this alteration influence the persistence and spatial distribution of the population? To explore these inquiries, we study the following nonlocal dispersal model:
where all the biological explanations for all parameters are the same as in the previous model (1).
In the article, unless stated otherwise, we will suppose that
For
For
It should be noted that there exist many difficulties in the nonlocal dispersal cases. One of the most important reasons is that principal eigenvalue might not exist and hence, it further constrains our research on assessing the local stability through linearized analysis. The second important reason is that the compactness of solutions orbits is a natural result in the random dispersal model, while this condition is not met in the
For the sake of clarity, set
in
where
Fortunately, due to the classical abstract monotone theory, system (3) with
Theorem 1.1
Assume that
If
If
Resorting to the above solution structure of Theorem 1.1 and verifying either the absence or uniqueness of positive steady states is determined directly by the properties of nonlocal operators and through arguments involving contradiction, we have the following important global result.
Theorem 1.2
For all nonnegative and nontrivial initial conditions, the subsequent assertions regarding system (3)are valid.
Assume that
If
If
If
Assume that
If
If
If
In numerous species, dispersion encompasses both local migration and a minor fraction of long-distance migration. Refer [37] and references therein. A famous example is a genetic model featuring partial panmixia, in which the diffusion term integrates both local and nonlocal dispersal; here the nonlocal dispersal serves as an approximation for long-distance migration. Furthermore, to gain a clearer insight into the competitive benefits of various dispersal strategies, Kao et al. [30,31] studied the model in which one species moves solely through random walk, while the other employs a non-local dispersal strategy. Motivated by these work, we propose the following model with more general dispersal:
We note here that if
Using a method similar to that employed in the proof of Theorem 1.2, one can derive the following outcome regarding the global dynamics of system (5).
Theorem 1.3
For all nonnegative and nontrivial initial conditions, the subsequent assertions regarding system (5)are true.
Assume that
If
If
If
Assume that
If
If
If
The remainder of this study is structured as follows: Section 2 examines the nonlocal model for a single species and demonstrates the existence and uniqueness of a semi-trivial steady state of system (3). Section 3 is devoted to showing the main Theorems 1.1 and 1.2. In Section 4, we analyze system (5) and establish Theorem 1.3. We give a short conclusion in Section 5.
2 Single species models with nonlocal dispersal
In this section, we consider two single species models
and
where
In order to investigate the existence and uniqueness of positive solutions of (6), we need to examine the local stability of the zero solution, which is decided by the signs of
where
Choosing 1 as a test function, one finds that
Lemma 2.1
Problem (6) possesses a unique positive solution
Next, we demonstrate the existence and uniqueness of positive solutions for problem (7).
Lemma 2.2
Problem (7) possesses a unique positive solution
3 Two species competition model with nonlocal dispersal
For ease of reference, we express the
where
Let
which determines the local stability/instability of
To stress the dependence on parameters of
Lemma 3.1
Fixing all parameters except b and regarding
i.e.,
Fixing all parameters except r and regarding
Remark 3.1
We remark here that
We next introduce some definitions and basic properties which will be beneficial for demonstrating the main results.
Definition 3.1
We say
The following result follows from some standard arguments and [20, Proposition 2.2].
Lemma 3.2
Let
Similar to that of ([1], Lemmas 2.3 and 2.5), we reach the following result.
Lemma 3.3
The subsequent statements are true.
If
There is a positive real number
where
We note here that similar to Lemma 2.2, one can obtain the existence and uniqueness of
Proof
Here we use a similar method as in the proof of [1, Lemma 2.3] and [20, Proposition 2.6]. For any
exists associated with the eigenfunction
Obviously, we see that
On the other hand, for small
Thus,
(ii) Next we turn to show that
Obviously, for small
On the other hand, we also have
Thus,
Lemma 3.4
If there is a coexistence steady state
where
Proof
The results follow directly from the approach of the upper and lower solutions method and maximum principle.□
Based on the above preparations, we can characterize the global dynamics of the competition model (3) with
Lemma 3.5
For all nonnegative and nontrivial initial conditions, the subsequent two statements are true.
If
Proof
For ease of reference, we will use
Then, applying the comparison principle to the first equation in system (3), one can derive that there is a constant
Next we show statement (i) is valid. Note that
Using the comparison principle for monotone systems, we obtain
According to Lemma 3.2, it can be concluded that, in the context of competitive order,
Furthermore, if the positive steady state is unique, one has
In view of Theorem 7.13 [14], one obtains
and
Combining with (13), we see that
Now, we turn to show (ii). From (11), (12), and Lemma 3.3 (ii), it can be concluded that system (3) possesses an upper solution
Again resorting to the comparison principle, we have
Note the assumption that system (3) has no positive steady states, this along with Lemma 3.2 suggests that
By (14), one has
Lemma 3.6
If
Proof
First of all, define
For any given
where
Then,
Clearly, for any
Thus,
This suggests that if
where we have used (15). Employing the implicit function theorem,
The following result is crucial for understanding the global dynamics. Our reasoning depends on examining the properties of nonlocal operators and certain integral relationships inspired by [2,40].
Proposition 3.1
Assume that
Proof
It is enough to demonstrate that if system (3) has two positive steady states
where
If
Using the first equation of system (3) satisfied by
and thus,
We will show that
In view of the left-hand side of (17), one has
By exchanging the position of variables
This implies
Similarly, by applying (16) and the second equation of system (3), we have
which gives that
Similar to the previous analysis and noting the fact that
Therefore, we have
By multiplying the second equation of (21) by
where we have used
which contradicts
3.1 Proof of Theorems 1.1 and 1.2
In this section, we establish the primary theorems.
Proof of Theorem 1.1
Statement (i) is a direct consequence of Lemmas 3.5, 3.6, and Proposition 3.1.
Next, we prove statement (ii). Based on Lemma 3.5 (ii), to demonstrate that
Let
Using the same arguments to derive (20), one has
Recalling that
Thus, we derive two inequalities as follows:
Multiplying the first equation of (22) by
where we have used Lemma 3.4. This is impossible. Therefore, system (3) does not possess positive steady state and hence,
Proof of Theorem 1.2
We will begin by demonstrating statement (i). Due to the definitions of
(i1) If
(i2) If
(i3) If
we conclude that
Using Lemma 3.4 and
Similar to Lemma 3.1 (monotonicity), one can show that
which contradicts (4). Therefore,
By applying arguments analogous to those used in proving statement (i), it can be shown that statement (ii) holds, which concludes the proof.□
4 Two species competition model with more general dispersal
In this section, we explore the global dynamics of (5). For
Then,
Similar to the arguments as that in Section 2, due to the assumption of
Proposition 4.1
There exists a positive critical value
i.e.,
Parallel results to Lemmas 3.2,3.3,3.4, and 3.5 can be established similarly. For simplicity of presentation, we do not state here. If system (5) admits a positive solution
To obtain the global dynamics of system (5), we first establish the key result similar to Proposition 3.1.
Proposition 4.2
Assume that
Proof
If it is not true, i.e., system (5) admits two positive steady states
Resorting to Proposition (3.1), one may assume that
Let
hold. Note that
Clearly, the other inequality in (25) can be handled similarly. Multiplying the second equation of (25) by
where we also have used
which contradicts
Proof of Theorem 1.3
By employing arguments to those used in the proof of Theorem 1.2, it can be demonstrated that Theorem 1.3 holds and we omit the details.□
5 Conclusion
In this study, we put forward new nonlocal diffusion models (3) and (5) and characterize the global dynamics of them. Model (3) comprises two nonlocal diffusion equations: one regulates the population’s dispersal and growth affected by toxicants, while the other outlines the dispersal, input, and decay of the toxicant. In the study process, we face many new difficulties in the nonlocal dispersal cases, for example, the local stability analysis cannot be used by primary eigenvalue analysis like the local diffusion case, and the nonlocal dispersal is lack of regularity such that the compactness of solutions orbits is not a natural result, and the reaction terms have distinct forms and are asymmetric, and so on. Based on some special skills, the linear stability analysis of the semi-trivial solution, the methods of lower-upper solutions, implicit function theorem, and the abstract monotone dynamical theory, we have overcome these difficulties and explore the existence and stability of steady states, which yields sufficient conditions that lead to population persistence or extinction. Theorems 1.2 and 1.3 describe the global dynamics of population-toxicant models (3) and (5) with the nonlocal dispersal operator and even the mixed dispersal strategies, respectively. Of course, these conclusions also apply to local diffusion model (1) when the advection rates
When our research model is limited to local dispersal, i.e.,
which is smaller than the critical value
Claim:
Lemma 5.1
For any
Proof
We first show that statement (i) is true. Let
Case I: If
Case II: If
Case III: If
For any
If
By a simple computation, we conclude that
Therefore, the proof is complete.□
Due to Lemma 5.1, we conclude that
if
if
other cases,
This suggests that we have successfully improved the critical value of [40, Lemma 3.9] when
However, there still exists some in-depth problems to the further study of the present work, at certain levels. First, some important results such as Theorem 1.2 (resp. Theorem 1.3) indeed gives the global dynamics of system (3) (resp. (5)) when
Acknowledgments
We are deeply grateful to the referees for carefully reading our paper and giving us various corrections and insightful suggestions/comments, which greatly improve the precision of our results and exposition of our manuscript.
-
Funding information: Ma’s research was supported by the National Natural Science Foundation of China (Nos. 12161003, 12361099, and 12301269), Guangdong Basic and Applied Basic Research Foundation (No. 2025A1515012064), and Guangdong Provincial Department of Education Innovation Team Project (No. 2023KCXTD063). Tang’s research was supported by Guangdong Basic and Applied Basic Research Foundation (Nos. 2023A1515012553, 2025A1515012045) and National Natural Science Foundation of China (No. 11901596).
-
Author contributions: All authors contributed equally to this work.
-
Conflict of interest: The authors state no conflict of interest.
-
Data availability statement: Data sharing is not applicable to this article as no new data were created or analyzed in this study.
References
[1] X. Bai and F. Li, Global dynamics of a competition model with nonlocal dispersal II: The full system, J. Differential Equations 258 (2015), 2655–2685, https://doi.org/10.1016/j.jde.2014.12.014. Search in Google Scholar
[2] X. Bai and F. Li, Classification of global dynamics of competition models with nonlocal dispersals I: symmetric kernels, Calc. Var. Partial Differential Equations 57 (2018), 1–35, https://doi.org/10.48550/arXiv.1704.02728. Search in Google Scholar
[3] H. Berestycki, J. Coville, and H. Vo, Persistence criteria for populations with non-local dispersion, J. Math. Bio. 72 (2016), 1693–1745, https://doi.org/10.1007/s00285-015-0911-2. Search in Google Scholar PubMed
[4] M. L. Cain, B. G. Milligan, and A. E. Strand, Long-distance seed dispersal in plant populations, Am. J. Bot. 87 (2000), 1217–1227, https://doi.org/10.2307/2656714. Search in Google Scholar
[5] R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-diffusion Equations. John Wiley & Sons, Ltd., Chichester, 2003, 10.1002/0470871296. Search in Google Scholar
[6] J. S. Clark, Why trees migrate so fast: confronting theory with dispersal biology and the paleorecord, Am. Nat. 152 (1998), 204–224, 10.1086/286162. Search in Google Scholar PubMed
[7] J. S. Clark, C. Fastie, G. Hurtt, S. T. Jackson, C. Johnson, G. A. King, et al., Reids paradox of rapid plant migration, BioScience 48 (1998), 13–24, https://doi.org/10.2307/1313224. Search in Google Scholar
[8] J. Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators, J. Differential Equations, 249 (2010), 2921–2953, https://doi.org/10.1016/j.jde.2010.07.003. Search in Google Scholar
[9] X. Deng, Q. Huang, and Z.-A. Wang, Global dynamics and pattern formation in a diffusive population-toxicant model with negative toxicant-taxis, SIAM J. Appl. Math. 83 (2023), no. 6, 2212–2236, https://doi.org/10.1137/22M1510881. Search in Google Scholar
[10] J. Dockery, V. Hutson, K. Mischaikow, and M. Pernarowski, The evolution of slow dispersal rates: A reaction-diffusion model, J. Math. Biol. 37 (1998), 61–83, https://doi.org/10.1007/s002850050120. Search in Google Scholar
[11] R. A. Erickson, S. B. Cox, J. L. Oates, T. A. Anderson, C. J. Salice, and K. R. Long, A Daphnia population model that considers pesticide exposure and demography stochasticity, Ecol. Model. 275 (2014), 37–47, https://doi.org/10.1016/j.ecolmodel.2013.12.015. Search in Google Scholar
[12] P. C. Fife, An integrodifferential analog of semilinear parabolic PDEs, Partial differential equations and applications, Lecture Notes in Pure and Appl. Math., Dekker, New York, vol. 177, 1996, pp. 137–145, https://doi.org/10.3109/14659899609084991.Search in Google Scholar
[13] H. I. Freedman and J. B. Shukla, Models for the effect of toxicant in single-species and predator-prey systems, J. Math. Biol. 30 (1991), 15–30, https://doi.org/10.1007/BF00168004. Search in Google Scholar PubMed
[14] M. Grinfeld, G. Hines, V. Hutson, K. Mischaikow, and G. T. Vickers, Non-local dispersal, Differ. Integral Equ. 18 (2005), no. 11, 1299–1320, 10.57262/die/1356059743. Search in Google Scholar
[15] T. G. Hallam, C. E. Clark, and G. S. Jordan, Effects of toxicants on populations: Aqualitative approach II. First order kinetics, J. Math. Biol. 18 (1983), 25–37, 10.1007/BF00275908. Search in Google Scholar PubMed
[16] T. G. Hallam, C. E. Clark, and R. R. Lassiter, Effects of toxicants on populations: A qualitative approach I. Equilibrium environmental exposure, Ecol. Model. 18 (1983), 291–304, https://doi.org/10.1016/0304-3800(83)90019-4. Search in Google Scholar
[17] T. G. Hallam and J. T. de Luna, Effects of toxicants on populations: A qualitative approach. III. Environmental and food chain pathways, J. Theor. Biol. 109 (1984), 411–429, https://doi.org/10.1016/S0022-5193(84)80090-9. Search in Google Scholar
[18] W. Hao, K.-Y. Lam, and Y. Lou, Ecological and evolutionary dynamics in advective environments: Critical domain size and boundary conditions, Discrete Contin. Dyn. Syst. Ser. B 26 (2021), 367–400, 10.3934/dcdsb.2020283. Search in Google Scholar
[19] X. He and W.-M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system: Diffusion and spatial heterogeneity I, Comm. Pure. Appl. Math. 69 (2016), 981–1014, https://doi.org/10.1002/cpa.21596. Search in Google Scholar
[20] G. Hetzer, T. Nguyen, and W. Shen, Coexistence and extinction in the Volterra-Lotka competition model with nonlocal dispersal, Commun. Pure Appl. Anal 11 (2012), 1699–1722, 10.3934/cpaa.2012.11.1699. Search in Google Scholar
[21] S. B. Hsu, H. L. Smith, and P. Waltman, Competitive exclusion and coexistence for competitive systems on ordered Banach spaces, Trans. Amer. Math. Soc. 348 (1996), no. 10, 4083–4094, 10.1090/S0002-9947-96-01724-2. Search in Google Scholar
[22] Q. Huang, G. Seo, and C. Shan, Bifurcations and global dynamics in a toxin-dependent aquatic population model, Math. Biosci. 296 (2018), 26–35, https://doi.org/10.1016/j.mbs.2017.11.013. Search in Google Scholar PubMed
[23] Q. Huang, H. Wang, and M. A. Lewis, The impact of environmental toxins on predator-prey dynamics, J. Theor. Biol. 378 (2015), 12–30, https://doi.org/10.1016/j.jtbi.2015.04.019. Search in Google Scholar PubMed
[24] V. Huston, S. Martinez, K. Miscaikow, and G. T. Vichers, The evolution of dispersal, J. Math. Biol. 47 (2003), 483–517, https://doi.org/10.1007/s00285-003-0210-1. Search in Google Scholar PubMed
[25] K.-Y. Lam, Y. Lou, and F. Lutscher, Evolution of dispersal in closed advective environments, J. Biol. Dyn. 14 (2014), 188–212, https://doi.org/10.1080/17513758.2014.969336. Search in Google Scholar PubMed
[26] F. Li, J. Coville, and X. Wang, On eigenvalue problems arising from nonlocal diffusion models, Discrete Contin. Dyn. Syst, 37 (2017), no. 2, 879–903, 10.3934/dcds.2017036. Search in Google Scholar
[27] Y. Lou and F. Lutscher, Evolution of dispersal in open advective environments, J. Math. Biol. 69 (2014), 1319–1342, https://doi.org/10.1007/s00285-013-0730-2. Search in Google Scholar PubMed
[28] Y. Lou, X.-Q. Zhao, and P. Zhou, Global dynamics of a Lotka-Volterra competition-diffusion advection system in heterogeneous environments, J. Math. Pures Appl. 121 (2019), 47–82, https://doi.org/10.1016/j.matpur.2018.06.010. Search in Google Scholar
[29] F. Lutscher, E. Pachepsky, and M. A. Lewis, The effect of dispersal patterns on stream populations, SIAM Rev. 47 (2005), no. 4, 749–772, https://doi.org/10.1137/050636152. Search in Google Scholar
[30] C.-Y. Kao, Y. Lou, and W. Shen, Random dispersal vs. non-local dispersal, Discrete Contin. Dyn. Syst. Ser. B 26 (2010), 551–596, 10.3934/dcds.2010.26.551. Search in Google Scholar
[31] C.-Y. Kao, Y. Lou, and W. Shen, Evolution of mixed dispersal in periodic environments, Discrete Contin. Dyn. Syst. Ser. B 17 (2012), 2047–2072, 10.3934/dcdsb.2012.17.2047. Search in Google Scholar
[32] A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern Perspectives Interdisciplinary Applied Mathematics, Vol. 14, 2nd ed. Springer, Berlin, 2001, https://doi.org/10.1007/978-1-4757-4978-6.Search in Google Scholar
[33] J. G. Skellam, Random dispersal in theoretical populations, Biometrika 38 (1951), 196–218, https://doi.org/10.2307/2332328.Search in Google Scholar
[34] J. A. Spromberg and W. J. Birge, Modeling the effects of chronic toxicity on fish populations: The influence of life-history strategies, Environ. Toxicol. Chem. 24 (2005), 1532–1540, 10.1897/04-160.1. Search in Google Scholar PubMed
[35] J. A. Spromberg and J. P. Meador, Relating chronic toxicity responses to population-level effects: A comparison of population-level parameters for three salmon species as a function of low-level toxicity, Ecol. Model. 199 (2006), 240–252, https://doi.org/10.1016/j.ecolmodel.2006.05.007. Search in Google Scholar
[36] R. A. Pastorok, S. M. Bartell, S. Ferson, and L. R. Ginzburg, Ecological Modeling in Risk Assessment: Chemical Effects on Populations, Ecosystems, and Landscapes, Lewis Publishers, Boca Raton, FL, 2001, https://doi.org/10.1201/9781420032321.Search in Google Scholar
[37] S. Petrovskii and A. Morozov, Dispersal in a statistically structured population: fat tails revisited, Am. Nat. 173 (2009), 278–289, https://doi.org/10.1086/595755. Search in Google Scholar PubMed
[38] D. Tang and Y. M. Chen, Global dynamics of a Lotka-Volterra competition-diffusion system in advective homogeneous environments, J. Differential Equations 269 (2020), 1465–1483, https://doi.org/10.1016/j.jde.2020.01.011. Search in Google Scholar
[39] P. Zhou, On a Lotka-Volterra competition system: diffusion vs advection, Calc. Var. Partial Differ. Equ. 55 (2016), 137, https://doi.org/10.1007/s00526-016-1082-8. Search in Google Scholar
[40] P. Zhou and Q. Huang, A spatiotemporal model for the effects of toxicants on populations in a polluted river, SIAM J. Appl. Math. 82 (2022), no. 1, 95–118, https://doi.org/10.1137/21M1405629. Search in Google Scholar
[41] P. Zhou, D. Tang, and D.-M. Xiao, On Lotka-Volterra competitive parabolic systems: exclusion, coexistence and bistability, J. Differential Equations 282 (2021), 596–625, https://doi.org/10.1016/j.jde.2021.02.031. Search in Google Scholar
© 2025 the author(s), published by De Gruyter
This work is licensed under the Creative Commons Attribution 4.0 International License.
Articles in the same Issue
- Research Articles
- Incompressible limit for the compressible viscoelastic fluids in critical space
- Concentrating solutions for double critical fractional Schrödinger-Poisson system with p-Laplacian in ℝ3
- Intervals of bifurcation points for semilinear elliptic problems
- On multiplicity of solutions to nonlinear Dirac equation with local super-quadratic growth
- Entire radial bounded solutions for Leray-Lions equations of (p, q)-type
- Combinatorial pth Calabi flows for total geodesic curvatures in spherical background geometry
- Sharp upper bounds for the capacity in the hyperbolic and Euclidean spaces
- Positive solutions for asymptotically linear Schrödinger equation on hyperbolic space
- Existence and non-degeneracy of the normalized spike solutions to the fractional Schrödinger equations
- Existence results for non-coercive problems
- Ground states for Schrödinger-Poisson system with zero mass and the Coulomb critical exponent
- Geometric characterization of generalized Hajłasz-Sobolev embedding domains
- Subharmonic solutions of first-order Hamiltonian systems
- Sharp asymptotic expansions of entire large solutions to a class of k-Hessian equations with weights
- Stability of pyramidal traveling fronts in time-periodic reaction–diffusion equations with degenerate monostable and ignition nonlinearities
- Ground-state solutions for fractional Kirchhoff-Choquard equations with critical growth
- Existence results of variable exponent double-phase multivalued elliptic inequalities with logarithmic perturbation and convections
- Homoclinic solutions in periodic partial difference equations
- Classical solution for compressible Navier-Stokes-Korteweg equations with zero sound speed
- Properties of minimizers for L2-subcritical Kirchhoff energy functionals
- Asymptotic behavior of global mild solutions to the Keller-Segel-Navier-Stokes system in Lorentz spaces
- Blow-up solutions to the Hartree-Fock type Schrödinger equation with the critical rotational speed
- Qualitative properties of solutions for dual fractional parabolic equations involving nonlocal Monge-Ampère operator
- Regularity for double-phase functionals with nearly linear growth and two modulating coefficients
- Uniform boundedness and compactness for the commutator of an extension of Riesz transform on stratified Lie groups
- Normalized solutions to nonlinear Schrödinger equations with mixed fractional Laplacians
- Existence of positive radial solutions of general quasilinear elliptic systems
- Low Mach number and non-resistive limit of magnetohydrodynamic equations with large temperature variations in general bounded domains
- Sharp viscous shock waves for relaxation model with degeneracy
- Bifurcation and multiplicity results for critical problems involving the p-Grushin operator
- Asymptotic behavior of solutions of a free boundary model with seasonal succession and impulsive harvesting
- Blowing-up solutions concentrated along minimal submanifolds for some supercritical Hamiltonian systems on Riemannian manifolds
- Stability of rarefaction wave for relaxed compressible Navier-Stokes equations with density-dependent viscosity
- Singularity for the macroscopic production model with Chaplygin gas
- Global strong solution of compressible flow with spherically symmetric data and density-dependent viscosities
- Global dynamics of population-toxicant models with nonlocal dispersals
- α-Mean curvature flow of non-compact complete convex hypersurfaces and the evolution of level sets
- High-energy solutions for coupled Schrödinger systems with critical growth and lack of compactness
- On the structure and lifespan of smooth solutions for the two-dimensional hyperbolic geometric flow equation
- Well-posedness for physical vacuum free boundary problem of compressible Euler equations with time-dependent damping
- On the existence of solutions of infinite systems of Volterra-Hammerstein-Stieltjes integral equations
- Remark on the analyticity of the fractional Fokker-Planck equation
- Continuous dependence on initial data for damped fourth-order wave equation with strain term
- Unilateral problems for quasilinear operators with fractional Riesz gradients
- 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
Articles in the same Issue
- Research Articles
- Incompressible limit for the compressible viscoelastic fluids in critical space
- Concentrating solutions for double critical fractional Schrödinger-Poisson system with p-Laplacian in ℝ3
- Intervals of bifurcation points for semilinear elliptic problems
- On multiplicity of solutions to nonlinear Dirac equation with local super-quadratic growth
- Entire radial bounded solutions for Leray-Lions equations of (p, q)-type
- Combinatorial pth Calabi flows for total geodesic curvatures in spherical background geometry
- Sharp upper bounds for the capacity in the hyperbolic and Euclidean spaces
- Positive solutions for asymptotically linear Schrödinger equation on hyperbolic space
- Existence and non-degeneracy of the normalized spike solutions to the fractional Schrödinger equations
- Existence results for non-coercive problems
- Ground states for Schrödinger-Poisson system with zero mass and the Coulomb critical exponent
- Geometric characterization of generalized Hajłasz-Sobolev embedding domains
- Subharmonic solutions of first-order Hamiltonian systems
- Sharp asymptotic expansions of entire large solutions to a class of k-Hessian equations with weights
- Stability of pyramidal traveling fronts in time-periodic reaction–diffusion equations with degenerate monostable and ignition nonlinearities
- Ground-state solutions for fractional Kirchhoff-Choquard equations with critical growth
- Existence results of variable exponent double-phase multivalued elliptic inequalities with logarithmic perturbation and convections
- Homoclinic solutions in periodic partial difference equations
- Classical solution for compressible Navier-Stokes-Korteweg equations with zero sound speed
- Properties of minimizers for L2-subcritical Kirchhoff energy functionals
- Asymptotic behavior of global mild solutions to the Keller-Segel-Navier-Stokes system in Lorentz spaces
- Blow-up solutions to the Hartree-Fock type Schrödinger equation with the critical rotational speed
- Qualitative properties of solutions for dual fractional parabolic equations involving nonlocal Monge-Ampère operator
- Regularity for double-phase functionals with nearly linear growth and two modulating coefficients
- Uniform boundedness and compactness for the commutator of an extension of Riesz transform on stratified Lie groups
- Normalized solutions to nonlinear Schrödinger equations with mixed fractional Laplacians
- Existence of positive radial solutions of general quasilinear elliptic systems
- Low Mach number and non-resistive limit of magnetohydrodynamic equations with large temperature variations in general bounded domains
- Sharp viscous shock waves for relaxation model with degeneracy
- Bifurcation and multiplicity results for critical problems involving the p-Grushin operator
- Asymptotic behavior of solutions of a free boundary model with seasonal succession and impulsive harvesting
- Blowing-up solutions concentrated along minimal submanifolds for some supercritical Hamiltonian systems on Riemannian manifolds
- Stability of rarefaction wave for relaxed compressible Navier-Stokes equations with density-dependent viscosity
- Singularity for the macroscopic production model with Chaplygin gas
- Global strong solution of compressible flow with spherically symmetric data and density-dependent viscosities
- Global dynamics of population-toxicant models with nonlocal dispersals
- α-Mean curvature flow of non-compact complete convex hypersurfaces and the evolution of level sets
- High-energy solutions for coupled Schrödinger systems with critical growth and lack of compactness
- On the structure and lifespan of smooth solutions for the two-dimensional hyperbolic geometric flow equation
- Well-posedness for physical vacuum free boundary problem of compressible Euler equations with time-dependent damping
- On the existence of solutions of infinite systems of Volterra-Hammerstein-Stieltjes integral equations
- Remark on the analyticity of the fractional Fokker-Planck equation
- Continuous dependence on initial data for damped fourth-order wave equation with strain term
- Unilateral problems for quasilinear operators with fractional Riesz gradients
- 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
