A diffusive weak Allee effect model with U-shaped emigration and matrix hostility

Nalin Fonseka; Jerome Goddard II; Ratnasingham Shivaji; Byungjae Son

doi:10.3934/dcdsb.2020356

Previous Article
The spatially heterogeneous diffusive rabies model and its shadow system
DCDS-B Home
This Issue
Next Article
High-order numerical method for two-dimensional Riesz space fractional advection-dispersion equation

October 2021, 26(10): 5509-5517. doi: 10.3934/dcdsb.2020356

A diffusive weak Allee effect model with U-shaped emigration and matrix hostility

Nalin Fonseka ^1, , Jerome Goddard II ^2,, , Ratnasingham Shivaji ^1, and Byungjae Son ^3,

1.	Department of Mathematics and Statistics, University of North Carolina at Greensboro, Greensboro, NC 27412, USA
2.	Department of Mathematics, Auburn University Montgomery, Montgomery, AL 36124, USA
3.	Department of Mathematics and Statistics, University of Maine, Orono, ME 04469, USA

^* Corresponding author: Jerome Goddard II, jgoddard@aum.edu

Received July 2020 Revised October 2020 Published October 2021 Early access December 2020

Fund Project: The second author is supported by NSF grant DMS-1853372 and the third author is supported by NSF grant DMS-1853352

We study positive solutions to steady state reaction diffusion equations of the form:

$ \begin{equation*} \; \; \begin{matrix} -\Delta u = \lambda f(u);\; \Omega \\ \; \; \alpha(u)\frac{\partial u}{\partial \eta}+\gamma\sqrt{\lambda}[1-\alpha(u)]u = 0; \; \partial \Omega\end{matrix} \end{equation*} $

where

$ u $

is the population density,

$ f(u) = \frac{1}{a}u(u+a)(1-u) $

represents a weak Allee effect type growth of the population with

$ a\in (0,1) $

$ \alpha(u) $

is the probability of the population staying in the habitat

$ \Omega $

when it reaches the boundary, and positive parameters

$ \lambda $

and

$ \gamma $

represent the domain scaling and effective exterior matrix hostility, respectively. In particular, we analyze the case when

$ \alpha(s) = \frac{1}{[1+(A - s)^2 + \epsilon]} $

for all

$ s \in [0,1] $

, where

$ A\in (0,1) $

and

$ \epsilon\geq 0 $

. In this case

$ 1-\alpha(s) $

represents a U-shaped relationship between density and emigration. Existence, nonexistence, and multiplicity results for this model are established via the method of sub-super solutions.

Keywords: Weak Allee effect, patch-level Allee effect, reaction diffusion model, density dependent emigration, habitat fragmentation.

Mathematics Subject Classification: Primary: 35J25, 35J60; Secondary: 35J66.

Citation: Nalin Fonseka, Jerome Goddard II, Ratnasingham Shivaji, Byungjae Son. A diffusive weak Allee effect model with U-shaped emigration and matrix hostility. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5509-5517. doi: 10.3934/dcdsb.2020356

References:

[1]	H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976), 620-709. doi: 10.1137/1018114.
[2]	R. S. Cantrell and C. Cosner, Density dependent behavior at habitat boundaries and the Allee effect, Bull. Math. Biol., 69 (2007), 2339-2360. doi: 10.1007/s11538-007-9222-0.
[3]	J. T. Cronin, N. Fonseka, J. Goddard II, J. Leonard and R. Shivaji, Modeling the effects of density dependent emigration, weak Allee effects, and matrix hostility on patch-level population persistence, Math. Biosci. Eng., 17 (2020), 1718-1742. doi: 10.3934/mbe.2020090.
[4]	J. T. Cronin, J. Goddard II and and R. Shivaji, Effects of patch matrix-composition and individual movement response on population persistence at the patch-level, Bull. Math. Biol., 81 (2019), 3933-3975. doi: 10.1007/s11538-019-00634-9.
[5]	N. Fonseka, J. Goddard, Q. Morris, R. Shivaji and B. Son, On the effects of the exterior matrix hostility and a U-shaped density dependent dispersal on a diffusive logistic growth model, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 3401-3415. doi: 10.3934/dcdss.2020245.
[6]	N. Fonseka, A. Muthunayake, R. Shivaji and B. Son, Singular reaction diffusion equations where a parameter influences the reaction term and the boundary condition, Topol. Methods Nonlinear Anal., Accepted.
[7]	J. Goddard II, Q. Morris, C. Payne and R. Shivaji, A diffusive logistic equation with U-shaped density dependent dispersal on the boundary, Topol. Methods Nonlinear Anal., 53 (2019), 335-349. doi: 10.12775/tmna.2018.047.
[8]	J. Goddard II, Q. A. Morris, S. B. Robinson and R. Shivaji, An exact bifurcation diagram for a reaction diffusion equation arising in population dynamics, Bound. Value Probl., 2018 (2018), Paper No. 170, 17 pp. doi: 10.1186/s13661-018-1090-z.
[9]	J. Goddard II and R. Shivaji, Stability analysis for positive solutions for classes of semilinear elliptic boundary-value problems with nonlinear boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 147 (2017), 1019-1040. doi: 10.1017/S0308210516000408.
[10]	R. R. Harman, J. Goddard II, R. Shivaji and J. T. Cronin, Frequency of occurrence and population-dynamic consequences of different forms of density-dependent emigration, Am. Nat., 195 (2019), 851-867. doi: 10.1086/708156.
[11]	F. Inkmann, Existence and multiplicity theorems for semilinear elliptic equations with nonlinear boundary conditions, Indiana Univ. Math. J., 31 (1982), 213-221. doi: 10.1512/iumj.1982.31.31019.
[12]	M. A. Rivas and S. B. Robinson, Eigencurves for linear elliptic equations, ESAIM Control Optim. Calc. Var., 25 (2019), Paper No. 45, 25 pp. doi: 10.1051/cocv/2018039.
[13]	D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 21 (1971/72), 979-1000. doi: 10.1512/iumj.1972.21.21079.
[14]	J. Shi and R. Shivaji, Persistence in reaction diffusion models with weak Allee effect, J. Math. Biol., 52 (2006), 807-829. doi: 10.1007/s00285-006-0373-7.
[15]	R. Shivaji, A remark on the existence of three solutions via sub-super solutions, Nonlinear Analysis and Applications, Lecture notes in pure and applied mathematics, 109 (1987), 561–566, Ed. V. Lakshmikantham.

show all references

References:

[1]	H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976), 620-709. doi: 10.1137/1018114.
[2]	R. S. Cantrell and C. Cosner, Density dependent behavior at habitat boundaries and the Allee effect, Bull. Math. Biol., 69 (2007), 2339-2360. doi: 10.1007/s11538-007-9222-0.
[3]	J. T. Cronin, N. Fonseka, J. Goddard II, J. Leonard and R. Shivaji, Modeling the effects of density dependent emigration, weak Allee effects, and matrix hostility on patch-level population persistence, Math. Biosci. Eng., 17 (2020), 1718-1742. doi: 10.3934/mbe.2020090.
[4]	J. T. Cronin, J. Goddard II and and R. Shivaji, Effects of patch matrix-composition and individual movement response on population persistence at the patch-level, Bull. Math. Biol., 81 (2019), 3933-3975. doi: 10.1007/s11538-019-00634-9.
[5]	N. Fonseka, J. Goddard, Q. Morris, R. Shivaji and B. Son, On the effects of the exterior matrix hostility and a U-shaped density dependent dispersal on a diffusive logistic growth model, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 3401-3415. doi: 10.3934/dcdss.2020245.
[6]	N. Fonseka, A. Muthunayake, R. Shivaji and B. Son, Singular reaction diffusion equations where a parameter influences the reaction term and the boundary condition, Topol. Methods Nonlinear Anal., Accepted.
[7]	J. Goddard II, Q. Morris, C. Payne and R. Shivaji, A diffusive logistic equation with U-shaped density dependent dispersal on the boundary, Topol. Methods Nonlinear Anal., 53 (2019), 335-349. doi: 10.12775/tmna.2018.047.
[8]	J. Goddard II, Q. A. Morris, S. B. Robinson and R. Shivaji, An exact bifurcation diagram for a reaction diffusion equation arising in population dynamics, Bound. Value Probl., 2018 (2018), Paper No. 170, 17 pp. doi: 10.1186/s13661-018-1090-z.
[9]	J. Goddard II and R. Shivaji, Stability analysis for positive solutions for classes of semilinear elliptic boundary-value problems with nonlinear boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 147 (2017), 1019-1040. doi: 10.1017/S0308210516000408.
[10]	R. R. Harman, J. Goddard II, R. Shivaji and J. T. Cronin, Frequency of occurrence and population-dynamic consequences of different forms of density-dependent emigration, Am. Nat., 195 (2019), 851-867. doi: 10.1086/708156.
[11]	F. Inkmann, Existence and multiplicity theorems for semilinear elliptic equations with nonlinear boundary conditions, Indiana Univ. Math. J., 31 (1982), 213-221. doi: 10.1512/iumj.1982.31.31019.
[12]	M. A. Rivas and S. B. Robinson, Eigencurves for linear elliptic equations, ESAIM Control Optim. Calc. Var., 25 (2019), Paper No. 45, 25 pp. doi: 10.1051/cocv/2018039.
[13]	D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 21 (1971/72), 979-1000. doi: 10.1512/iumj.1972.21.21079.
[14]	J. Shi and R. Shivaji, Persistence in reaction diffusion models with weak Allee effect, J. Math. Biol., 52 (2006), 807-829. doi: 10.1007/s00285-006-0373-7.
[15]	R. Shivaji, A remark on the existence of three solutions via sub-super solutions, Nonlinear Analysis and Applications, Lecture notes in pure and applied mathematics, 109 (1987), 561–566, Ed. V. Lakshmikantham.

Figure 1. Illustration of $ 1-\alpha(s) $ and $ f(s) $

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2. Bifurcation diagram for the solution set of (3)

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3. Bifurcation diagram for the solution set of (3) for $ \gamma \gg 1 $ and $ \epsilon \approx 0 $.

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4. Graphs of $ \kappa $ vs $ B_1(\kappa) $ and $ \frac{\kappa^2}{\gamma^2 (g(0))^2} $

Figure Options

Download full-size image

Download as PowerPoint slide

[1]	Jim M. Cushing. The evolutionary dynamics of a population model with a strong Allee effect. Mathematical Biosciences & Engineering, 2015, 12 (4) : 643-660. doi: 10.3934/mbe.2015.12.643
[2]	Dianmo Li, Zhen Zhang, Zufei Ma, Baoyu Xie, Rui Wang. Allee effect and a catastrophe model of population dynamics. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 629-634. doi: 10.3934/dcdsb.2004.4.629
[3]	Elena Braverman, Alexandra Rodkina. Stochastic difference equations with the Allee effect. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 5929-5949. doi: 10.3934/dcds.2016060
[4]	Miljana JovanoviĆ, Marija KrstiĆ. Extinction in stochastic predator-prey population model with Allee effect on prey. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2651-2667. doi: 10.3934/dcdsb.2017129
[5]	Chuang Xu. Strong Allee effect in a stochastic logistic model with mate limitation and stochastic immigration. Discrete and Continuous Dynamical Systems - B, 2016, 21 (7) : 2321-2336. doi: 10.3934/dcdsb.2016049
[6]	Pengmiao Hao, Xuechen Wang, Junjie Wei. Global Hopf bifurcation of a population model with stage structure and strong Allee effect. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 973-993. doi: 10.3934/dcdss.2017051
[7]	Liang Zhao, Jianhe Shen. Canards and homoclinic orbits in a slow-fast modified May-Holling-Tanner predator-prey model with weak multiple Allee effect. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022018
[8]	Yujing Gao, Bingtuan Li. Dynamics of a ratio-dependent predator-prey system with a strong Allee effect. Discrete and Continuous Dynamical Systems - B, 2013, 18 (9) : 2283-2313. doi: 10.3934/dcdsb.2013.18.2283
[9]	Wenjie Ni, Mingxin Wang. Dynamical properties of a Leslie-Gower prey-predator model with strong Allee effect in prey. Discrete and Continuous Dynamical Systems - B, 2017, 22 (9) : 3409-3420. doi: 10.3934/dcdsb.2017172
[10]	Moitri Sen, Malay Banerjee, Yasuhiro Takeuchi. Influence of Allee effect in prey populations on the dynamics of two-prey-one-predator model. Mathematical Biosciences & Engineering, 2018, 15 (4) : 883-904. doi: 10.3934/mbe.2018040
[11]	Qizhen Xiao, Binxiang Dai. Heteroclinic bifurcation for a general predator-prey model with Allee effect and state feedback impulsive control strategy. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1065-1081. doi: 10.3934/mbe.2015.12.1065
[12]	Nika Lazaryan, Hassan Sedaghat. Extinction and the Allee effect in an age structured Ricker population model with inter-stage interaction. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 731-747. doi: 10.3934/dcdsb.2018040
[13]	Yi Yang, Robert J. Sacker. Periodic unimodal Allee maps, the semigroup property and the $\lambda$-Ricker map with Allee effect. Discrete and Continuous Dynamical Systems - B, 2014, 19 (2) : 589-606. doi: 10.3934/dcdsb.2014.19.589
[14]	Wei Feng, Michael Freeze, Xin Lu. On competition models under allee effect: Asymptotic behavior and traveling waves. Communications on Pure and Applied Analysis, 2020, 19 (12) : 5609-5626. doi: 10.3934/cpaa.2020256
[15]	Xiaoyuan Chang, Junping Shi. Bistable and oscillatory dynamics of Nicholson's blowflies equation with Allee effect. Discrete and Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021242
[16]	Eduardo González-Olivares, Betsabé González-Yañez, Jaime Mena-Lorca, José D. Flores. Uniqueness of limit cycles and multiple attractors in a Gause-type predator-prey model with nonmonotonic functional response and Allee effect on prey. Mathematical Biosciences & Engineering, 2013, 10 (2) : 345-367. doi: 10.3934/mbe.2013.10.345
[17]	Na Min, Mingxin Wang. Hopf bifurcation and steady-state bifurcation for a Leslie-Gower prey-predator model with strong Allee effect in prey. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 1071-1099. doi: 10.3934/dcds.2019045
[18]	Yongli Cai, Malay Banerjee, Yun Kang, Weiming Wang. Spatiotemporal complexity in a predator--prey model with weak Allee effects. Mathematical Biosciences & Engineering, 2014, 11 (6) : 1247-1274. doi: 10.3934/mbe.2014.11.1247
[19]	Meng Zhao, Wantong Li, Yihong Du. The effect of nonlocal reaction in an epidemic model with nonlocal diffusion and free boundaries. Communications on Pure and Applied Analysis, 2020, 19 (9) : 4599-4620. doi: 10.3934/cpaa.2020208
[20]	J. Leonel Rocha, Abdel-Kaddous Taha, Danièle Fournier-Prunaret. Explosion birth and extinction: Double big bang bifurcations and Allee effect in Tsoularis-Wallace's growth models. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 3131-3163. doi: 10.3934/dcdsb.2015.20.3131

2020 Impact Factor: 1.327

Tools

Metrics

Tools

Article outline

Figures and Tables

2020 Impact Factor: 1.327

Tools

Figures and Tables

2020 Impact Factor: 1.327

American Institute of Mathematical Sciences

A diffusive weak Allee effect model with U-shaped emigration and matrix hostility

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

A diffusive weak Allee effect model with U-shaped emigration and matrix hostility

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors