Spatial Dynamics of a Leslie–Gower Type Predator-Prey Model with Interval Parameters

Wang, Caiyun; Guo, Min; Lan, Wangsen; Xu, Xiaoxin

doi:https://doi.org/10.1155/2022/2483688

Discrete Dynamics in Nature and Society

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Abstract Introduction Discussion and Conclusions Appendix Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 2483688 | https://doi.org/10.1155/2022/2483688

Spatial Dynamics of a Leslie–Gower Type Predator-Prey Model with Interval Parameters

Caiyun Wang,^1,2,3Min Guo,^1,2Wangsen Lan,³and Xiaoxin Xu⁴

Academic Editor: Binxiang Dai

Received09 Oct 2022

Revised12 Nov 2022

Accepted14 Nov 2022

Published12 Dec 2022

Abstract

Due to various imprecisions in nature, imprecise parameters in biological modeling should be taken into account. This paper studies the spatial dynamics of an imprecise prey-predator model of the Leslie–Gower type by presenting imprecise parameters as interval parameters. First, conditions of Turing instability are obtained via bifurcation analysis and interval-valued functions. Then, the effects of interval parameters on pattern selection are discussed via multiple-scale analysis. We discover that when all the parameters of the model are interval parameters, the value of the controlled parameter increases, and the range of the pattern selection domain expands as the value of the interval variable increases, i.e., both the controlled parameter and boundary of the pattern selection domain are interval numbers. Finally, under the effects of the interval parameters of diffusion and the prey’s conversion rate into biomass for the predator, the density of the prey decreases or increases, respectively, and the structure or the microstructure of the pattern of the model changes with the growing value of the interval variable. This paper provides a new perspective on the study of the spatial predator-prey model.

1. Introduction

The increasing interest in mathematical modeling of ecosystems has heightened the need for understanding the dynamic behaviors of species. Of particular interest are the spatial dynamics of the prey-predator model, one of which is the following mite model introduced by Wollkind et al. [1, 2], whose work was based on the following [3]:

The predator’s and prey’s population densities are denoted by and , where are all positive parameters. is referred to as the intrinsic growth rate of the prey without predation. is the intrinsic growth rate of the predator. The feature of this model is that the prey’s growth rate is logistic growth with capacity . Yet, the predator’s growth rate is interpreted as “logistic growth” with “carrying capacity” , which is called the Leslie–Gower term [4, 5]. estimates the decrease in predator population density caused by the lack (per capita ) of its favorite prey, where refers to the conversion rate into the predator’s biomass. Although the predator can convert to the substitute of its preferred resource, its population density will suffer due to the availability of its favorite food . , termed as the functional response with the meaning of prey consumption rate of per predator at prey density and predator density , can be chosen as Holling type I–III functions [6–9]. Particularly, Holling type III function is more suitable for vertebrate species, i.e., , where and are referred to as half-saturation constant and capture rate. The process that the predators are able to change their predation rate and convert from less common to more common prey is the learning process. The learning process can make the foraging system more efficient because the predators can switch to alternative food resources when they know that their population fluctuates. Holling III functional response is for this situation [10].

On the other hand, diffusion is one of the spatial motions in the prey-predator model, which depicts the random mobility of individual species. By incorporating spatial motion, model (1) becomes the spatial predator-prey model of the Leslie–Gower type. Although there are many papers about the spatial prey-predator model of the Leslie–Gower type, such as bifurcation analysis or global stability of steady states, and so on, we mainly discuss spatial patterns [11–15]. There have been plenty of studies about spatial patterns of this model [16–22]. For the spatial prey-predator model of the Leslie–Gower type, some researchers focused on the Holling–Tanner type, where is chosen as Holling type II response function. They mathematically analyzed the Turing instability of the model and numerically simulated different kinds of patterns [23, 24]. Some papers discussed specific effects on pattern formation of the model, such as cannibalism, delay effects, and Allee effect [25–27]. Other papers discussed the effects of the cross-diffusion [28–30].

Yet, diffusion rate in above papers mentioned is constant. In fact, dispersal or diffusion behavior among species is not simple movement, but is a complex process which can be normally divided into three stages: emigration, movement between areas, and immigration. Moreover, dispersal rates in the process of three stages are condition-dependent and are density-dependent in particular [31]. The density-dependent dispersal rate can be either positive or negative. The common species with positive or negative density-dependent dispersal rate are insects, birds or mammals in the real word [32, 33]. Then, based on the work above, researchers have studied spatial dynamics of diffusive system with varying diffusion rate [34, 35]. However, coefficients in above papers are accurate, and all the spatial predator-prey models mentioned above fail to view the biological parameters as imprecise parameters. In reality, the impreciseness can happen because of incomplete information during the determination of the initial value, measurement, and data collection. After Zadeh’s pioneering work of introducing uncertainty into mathematically modeling [36], more and more researchers have incorporated impreciseness in the mathematical models via fuzzy sets [37–39]. As far as predator-prey models are concerned, although there are researches about the imprecise ODE model or imprecise diffusive model [40–44], few papers are about patterns of spatial model with interval diffusion rate. Motivated by the work [34, 35, 44], we will study Turing patterns of an imprecise prey-predator model of the Leslie–Gower type with Holling type III function in this paper. For simplicity, we set up the corresponding accurate model first:where , a bounded domain, satisfies , and is the outward unit normal vector of the smooth bound of . The coefficients of diffusion of prey and predator are .

By introducing dimensionless variables , and , system (2) can be simplified aswhere and .

In Section 2, interval number and interval-valued function are introduced, and the imprecise model corresponding to model (3) is set up. Turing instability and amplitude equations are analyzed mathematically in Section 3. Theoretical analysis is illustrated by numerical simulations in Section 4. At last, conclusions about the model are drawn in Section 5.

2. Prerequisites and Modeling

In this section, we introduce interval numbers, interval-valued function, and arithmetic operations about it. Then, we set up the imprecise predator-prey model of the Leslie–Gower type.

2.1. Definitions

Definition 1. (see Reference [40]). An interval numberis presented by a closed interval. Letbe the set of real numbers and define, whereare the upper and lower limit of. Each real numbercan be denoted as interval number.
Let and be any two interval numbers, which define the following arithmetic operations:(i)Addition: (ii)Subtraction: (iii)Scalar multiplication: (iv)Multiplication: (v)Division:

Definition 2. (see [40]). An interval-valued function can be defined asfor any interval number, wherecalled the interval variable.

2.2. Imprecise Leslie–Gower Type Predator-Prey Model with Diffusion

Assume that parameters in system (3) are imprecise and presented as interval numbers. Let , and be the interval parameters corresponding to , and . We have the imprecise predator-prey model of the Leslie–Gower type with diffusion.where , , , , and . . Apparently, the interval-valued function for the interval is , which is a continuous and strictly increasing function. Based on Theorem 1 in [40], the imprecise form of model (4) is given by

3. Bifurcation Analysis

In this section, we analyze the existence of positive equilibrium of model (5) first and then find the conditions that ensure the appearance of the Turing pattern.

3.1. Analysis of Positive Equilibrium of Model (5)

The corresponding nondiffusion model of (5) is

Obviously, model (6) has an equilibrium . We are interested in the interior equilibrium points , where , and satisfieswhere and . The numbers of positive equilibria of (6) are the same as the numbers of positive real roots of in . According to [45], (7) has at most three roots and at least one root in the interval , so system (6) has at most three equilibria and at least one equilibrium.

Suppose is any positive equilibrium of model (6), at which the Jacobian matrix is calculated by

is stable when the following conditions hold:

Andwhere are trace and determinant of matrix given by

3.2. Analysis of Turing Instability of Model (5)

We obtain the conditions for Turing instability in this subsection. Suppose is one of the positive and steady equilibrium, whose characteristic polynomial iswhere the expression of iswhere is the wavenumber. The eigenvalue of (13) can be obtained by the following equation:wherewhere

Then, the eigenvalue can be expressed as

guarantees Turing instability. Clearly , the necessary condition for Turing instability is for some value of . That is,

And

As a result, if (9), (10), (18), and (19) hold, then Turing patterns of model (5) can appear.

3.3. Pattern Selection of Model (5)

In order to study the effects of interval parameters on pattern selection, we choose as controlled number to deduce amplitude equations of (5) via multiple-scale analysis.

Around , model (5) can be rewritten aswherewhere and are the amplitudes of the pattern of and y, c.c. denotes the complex conjugate item of , and

And

The detailed expression of derivatives of all orders of to can be seen (A.1)–(A.10) in Appendix.

Near Turing bifurcation threshold, , the bifurcation parameter , the variable , time , and the nonlinear term can be expanded aswhere

The linear operator is dissembled via Taylor expansion as follows:where

And

The derivation of the amplitude iswhere

Substitute (24)–(29) into (20) and collect , and Following linear systems can be given:where

And

Solving (30), the solution is as follows:where , denotes the complex conjugate item of and is the amplitude of the pattern with the mode under the first order perturbation, the form of which is determined by higher order term. Substituting equation (35) to (31), one can obtain the explicit expression of , and in (A.11) Appendix.

From the Fredholm solvability condition [13], we know that if the vector function of the right side of (31) is orthogonal to the eigenvectors of the zero eigenvalue of the adjoint operator of , which can be denoted by , there exists a nontrivial solution of (31). Substituting (35) into right side of (31), the eigenvector of the operator is given as follows:

In the light of the Fredholm solvability condition, the orthogonality condition is

Then, (37) yieldswhere is the conjugation of .

Let the form of the solution of (31) bewhere expressions of , and are given as (A.12)–(A.18) in Appendix.

Substituting (39) into (32) and collecting the coefficients of , and the expressions of , and can be given in (A.19) in Appendix.

Employing the Fredholm solvability condition in (32) again, we havewhere is the conjugation of .

Multiplying (38) and (40) by and , in consideration of (29) and the following amplitude equations are given:where is the conjugation of , andwhere the expression that , and stand for can be seen (A.20)–(A.24) in Appendix.

According to [13], we get the results of pattern selection in Table 1.

4. Numerical Simulations

In this part, we numerically simulated patterns of system (5) in 2-dimensional and space area, employing Euler-forward finite difference method and zero-flux boundary condition. We chose space step size as and time step size as with time interval . For the sake of similar patterns of both prey and predator, the prey’s pattern was chosen to investigate in detail, and the simulations will not stop until the patterns reach their steady state. Additionally, we varied the interval variable to show how the interval parameter would affect the spatial dynamics of system (5).

4.1. Effects of Interval Parameters on Controlled Parameter and Pattern Selection

In this section, we selected all parameters as interval parameters (Table 2) and studied the effects of interval parameters on controlled parameter and pattern selection.

It can be seen from Table 3 that the values of controlled parameter increase from 0.2860078311 to 0.3264191705 as the values of interval variable increase. According to the definition of interval number, it means that when parameters of system (5) are interval numbers, the controlled parameter is also an interval number with the value [0.2860078311, 0.3264191705]. By the same way, the boundary and of the domain of pattern selection (Table 1), where the controlled parameter lies, are also interval numbers. Moreover, the domains, i.e., and , of pattern selection expand as an interval variable increases. Figures 1–3 show the patterns that appear in corresponding pattern selection domain when , and . All the patterns in Figures 1–3 are in agreement with the theoretical analysis in Table 1, i.e., spot patterns (Figures 1(a), 2(a), and 3(a)) appear when lies in , and mixture of spots and stripes (Figures 1(b), 2(b), and 3(b)) emerge when lies in , and stripe patterns ((Figures 1(c), 2(c), and 2(c)) arise when . From this, we conclude that interval parameter of system (5) makes the boundary and be an interval number or changes the range of pattern selection domain and , but it does not change the types of patterns in the corresponding pattern selection domain.

(a)

(b)

(c)

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(b)

(c)

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(b)

(c)

4.2. Effects of Interval Parameter on Patterns of System (5)

The diffusion coefficient is an important parameter with the meaning of random motion, so we figured out what influences the impreciseness of diffusion would exert on system (5) by choosing values of parameters in Table 4, where we selected as interval parameters and others as accurate.

It can be seen from Figure 4 that the microstructure of spot pattern of the prey changes with the increasing of the interval variable : the density of the cold spot pattern gradually becomes sparse. Specifically, there are fewer and fewer isolated areas formed by the low density of the prey. To differentiate the patterns in Figure 4 more clearly, we plotted time-series of prey’s density at spatial location (21, 21) in Figure 5 and spatial mean density of the prey with respect to interval variable in Figure 6. From Figure 5, when , the density of prey lies in (0.4, 0.45) with in Figure 5(a), (0.35, 0.4) with in Figure 5(b), (0.3, 0.35) with in Figure 5(c), (0.24, 0.26) with in Figure 5(d), and (0.2, 0.21) with in Figure 5(e). One can see that the density of the prey decreases along with the increasing of , which is in accordance with results of Figure 6. From above analysis, it means that the interval parameter not only affects the microstructure of pattern of the prey but also the density of the prey.

(a)

(b)

(c)

(d)

(e)

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4.3. Effects of Interval Parameter on Patterns of System (5)

The parameter denotes the reliance of predators on prey, which is an intrinsic factor of predator-prey systems of the Leslie–Gower type, so we studied the effects of the interval parameter on model (5). For this purpose, we chose as the interval parameter and others as accurate, the values of which were provided in Table 5. Then, we simulated patterns of prey numerically.

From Figure 7, patterns of the prey transit orderly with the increasing of the interval variable : hot spot pattern (Figure 7(a)), mixture of hot spot and stripe pattern (Figure 7(b)), stripe pattern (Figure 7(c)), mixture of cold spot and stripe pattern (Figure 7(d)), cold spot pattern (Figure 7(e)) with the increasing of the interval variable of . It is easy to see the differences of patterns in Figure 7. Compared with Figure 4, Figures 8 and 9 show that the changing of prey’s density show an opposite trend, i.e., the density of the prey increases as the values of the interval variable grows. We conclude that the interval parameter has a remarkable effect on types of pattern and the density of the prey.

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5. Discussion and Conclusions

The spatial dynamics of a prey-predator model with an interval parameter were initially studied in this paper, compared with the former work [23–30]. First, conditions of Turing instability of the model with an interval parameter were gained via an interval-valued function and mathematical analysis. Then, the effects of all or some interval parameters on the spatial dynamics of system (5) were discussed. Specifically, via multiple-scale analysis and numerical simulation, pattern selection and pattern transitions for system (5) under interval parameters were obtained. It is well known that multiple-scale analysis is effective when the values of parameters are very close to the controlled parameter. According to our results, whatever the sensitive parameter, the controlled parameter is also an interval number. The results in Section 4.2 and 4.3 show that the effects of different interval parameters on system (5) are different. In this paper, the impreciseness is presented by the interval parameter, so we conclude that the impreciseness of the parameter does affect the behavior of the entire spatial biological system. Based on the conclusion, our hypothesis proposed in the introduction that researchers should incorporate impreciseness when they make mathematical modeling is reasonable.

It should be noted that we choose as a controlled number to study the effects of interval parameters on pattern selection. Other parameters can also be chosen as controlled parameters. Moreover, the interval variable describes any position of the value of an interval-valued function on the closed interval that presents the interval number. According to the definition of an interval-valued function, the bigger value of means the value of the interval-valued function lies closer to the upper limit of the interval number. Our results reveal the tendency of the spatial dynamics in the process of the values of an interval-valued function increasing to the upper limit of the interval number (see Tables 6 and 7). This study has concentrated on the effects of interval number on the spatial dynamics of a spatial prey-predator model. Despite its preliminary nature, imprecise parameters make the model closer to the reality of a lack of accurate information and provide a new perspective on studying spatial predator-prey models. In future work, it intrigues us to think about the effects of impreciseness on the spatial epidemic model.

Appendix

The detailed expressions in Subsection 3.3 are in the following:

Data Availability

Data and models generated or used during the study are included in the manuscript.

Conflicts of Interest

The authors declare that they have no conflicts interest.

Authors’ Contributions

Caiyun Wang and Min Guo conceived and designed the model; Wangsen Lan and Xiaoxin Xu performed the numerical simulations; Caiyun Wang and Min Guo analyzed the data; and Caiyun Wang, Min Guo, Wangsen Lan, and Xiaoxin Xu wrote the paper. All authors read and approved the final manuscript.

Acknowledgments

This work was supported by the research project supported by the Shanxi Scholarship Council of China (grant no. 2021-107) and Research on the Standardization and Innovation of Government Service in Shanxi Province (grant no. 2021JY0055).

References

D. J. Wollkind and J. A. Logan, “Temperature-dependent predator–prey mite ecosystem on apple tree foliage,” Journal of Mathematical Biology, vol. 6, no. 3, pp. 265–283, 1978.
View at: Publisher Site | Google Scholar
D. J. Wollkind, J. B. Collings, and J. A. Logan, “Metastability in a temperature-dependent model system for predator–prey mite outbreak interactions on fruit trees,” Bulletin of Mathematical Biology, vol. 50, no. 4, pp. 379–409, 1988.
View at: Publisher Site | Google Scholar
R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, NJ, USA, 1974.
P. H. Leslie, “Some further notes on the use of matrices in population mathematics,” Biometrika, vol. 35, no. 3/4, pp. 213–245, 1948.
View at: Publisher Site | Google Scholar
P. H. Leslie and J. C. Gower, “The properties of a stochastic model for the predator–prey type of interaction between two species,” Biometrika, vol. 47, no. 3/4, pp. 219–234, 1960.
View at: Publisher Site | Google Scholar
C. S. Holling, “The components of predation as revealed by a study of small-mammal predation of the european pine sawfly,” The Canadian Entomologist, vol. 91, no. 5, pp. 293–320, 1959.
View at: Publisher Site | Google Scholar
C. S. Holling, “Some characteristics of simple types of predation and parasitism,” The Canadian Entomologist, vol. 91, no. 7, pp. 385–398, 1959.
View at: Publisher Site | Google Scholar
C. S. Holling, “The functional response of predators to prey density and its role in mimicry and population regulation,” Memoirs of the Entomological Society of Canada, vol. 97, no. S45, pp. 5–60, 1965.
View at: Publisher Site | Google Scholar
C. S. Holling, “The functional response of invertebrate predators to prey density,” Memoirs of the Entomological Society of Canada, vol. 98, no. S48, pp. 5–86, 1966.
View at: Publisher Site | Google Scholar
L. A. Real, “The kinetics of functional response,” The American Naturalist, vol. 111, no. 978, pp. 289–300, 1977.
View at: Publisher Site | Google Scholar
W. Abid, R. Yafia, M. A. Aziz-Alaoui, and A. Aghriche, “Turing instability and Hopf bifurcation in a modified Leslie–Gower predator–prey model with cross-diffusion,” International Journal of Bifurcation and Chaos, vol. 28, no. 07, pp. 1850089–1850117, 2018.
View at: Publisher Site | Google Scholar
G. Sun, S. Sarwardi, P. J. Pal, and M. D. S. Rahman, “The spatial patterns through diffusion-driven instability in modified Leslie–Gower and Holling-type II predator–prey model,” Journal of Biological Systems, vol. 18, no. 03, pp. 593–603, 2010.
View at: Publisher Site | Google Scholar
Q. Ouyang, Pattern Formation in Reaction-Diffusion Systems, Shanghai Sci.-Tech, Education Publishing House, Shanghai, China, 2000.
A. M. Turing, “The chemical basis of morphogenesis,” Bulletin of Mathematical Biology, vol. 52, no. 1–2, pp. 153–197, 1990.
View at: Publisher Site | Google Scholar
S. Kondo and T. Miura, “Reaction-diffusion model as a framework for understanding biological pattern formation,” Science, vol. 329, no. 5999, pp. 1616–1620, 2010.
View at: Publisher Site | Google Scholar
H. Meinhardt, Models of Biological Pattern Formation, Academic Press, London, UK, 1982.
G. P. Hu, X. L. Li, and Y. P. Wang, “Pattern formation and spatiotemporal chaos in a reaction-diffusion predator–prey system,” Nonlinear Dynamics, vol. 81, no. 1-2, pp. 265–275, 2015.
View at: Publisher Site | Google Scholar
L. Zhang, F. Zhang, and S. Ruan, “Linear and weakly nonlinear stability analyses of Turing patterns for diffusive predator–prey systems in freshwater marsh landscapes,” Bulletin of Mathematical Biology, vol. 79, no. 3, pp. 560–593, 2017.
View at: Publisher Site | Google Scholar
H. B. Shi and S. Ruan, “Spatial temporal and spatiotemporal patterns of diffusive predator–prey models with mutual interference,” IMA Journal of Applied Mathematics, vol. 80, no. 5, pp. 1534–1568, 2015.
View at: Publisher Site | Google Scholar
S. Djilali and S. Bentout, “Spatiotemporal patterns in a diffusive predator–prey model with prey social behavior,” Acta Applicandae Mathematica, vol. 169, no. 1, pp. 125–143, 2020.
View at: Publisher Site | Google Scholar
X. Y. Gao, “Global solution and spatial patterns for a ratio-dependent predator–prey model with predator-taxis,” Results in Mathematics, vol. 77, no. 2, pp. 66–19, 2022.
View at: Publisher Site | Google Scholar
D. X. Geng, W. H. Jiang, Y. Lou, and H. B. Wang, “Spatiotemporal patterns in a diffusive predator–prey system with nonlocal intraspecific prey competition,” Studies in Applied Mathematics, vol. 148, no. 1, pp. 396–432, 2021.
View at: Publisher Site | Google Scholar
L. L. Chang, G. Q. Sun, Z. Wang, and Z. Jin, “Rich dynamics in a spatial predator–prey model with delay,” Applied Mathematics and Computation, vol. 256, pp. 540–550, 2015.
View at: Publisher Site | Google Scholar
P. P. Liu and Y. Xue, “Spatiotemporal dynamics of a predator–prey model,” Nonlinear Dynamics, vol. 69, no. 1–2, pp. 71–77, 2012.
View at: Publisher Site | Google Scholar
C. Y. Wang, L. L. Chang, and H. F. Liu, “Spatial patterns of a predator–prey system of Leslie Type with time delay,” PLoS One, vol. 11, no. 3, Article ID e0150503, 2016.
View at: Publisher Site | Google Scholar
G. Q. Sun, “Mathematical modeling of population dynamics with Allee effect,” Nonlinear Dynamics, vol. 85, no. 1, pp. 1–12, 2016.
View at: Publisher Site | Google Scholar
A. Basheer, E. Quansah, S. Bhowmick, and R. D. Parshad, “Prey cannibalism alters the dynamics of Holling–Tanner type predator–prey models,” Nonlinear Dynamics, vol. 85, no. 4, pp. 2549–2567, 2016.
View at: Publisher Site | Google Scholar
W. M. Wang, Z. G. Guo, R. K. Upadhyay, and Y. H. Lin, “Pattern Formation in a cross-diffusive holling–tanner model,” Discrete Dynamics in Nature and Society, vol. 2012, Article ID 828219, pp. 1–12, 2012.
View at: Publisher Site | Google Scholar
G. Q. Sun, Z. Jin, L. Li, M. Haque, and B. L. Li, “Spatial patterns of a predator–prey model with cross diffusion,” Nonlinear Dynamics, vol. 69, no. 4, pp. 1631–1638, 2012.
View at: Publisher Site | Google Scholar
R. Peng, M. Wang, and G. Yang, “Stationary patterns of the Holling–Tanner prey–predator model with diffusion and cross-diffusion,” Applied Mathematics and Computation, vol. 196, no. 2, pp. 570–577, 2008.
View at: Publisher Site | Google Scholar
D. E. Bowler and T. G. Benton, “Causes and consequences of animal dispersal strategies: relating individual behaviour to spatial dynamics,” Biological Reviews, vol. 80, no. 2, pp. 205–225, 2005.
View at: Publisher Site | Google Scholar
E. Matthysen, “Density-dependent dispersal in birds and mammals,” Ecography, vol. 28, no. 3, pp. 403–416, 2005.
View at: Publisher Site | Google Scholar
N. Cappucccino and P. W. Price, Population Dynamics: New Approaches and Synthesis, Academic Press, New York, USA, 1995.
M. J. Smith, J. A. Sherratt, and X. Lambin, “The effects of density-dependent dispersal on the spatiotemporal dynamics of cyclic populations,” Journal of Theoretical Biology, vol. 254, no. 2, pp. 264–274, 2008.
View at: Publisher Site | Google Scholar
M. Kumar and S. Abbas, “Diffusive size-structured population model with time-varying diffusion rate,” Discrete and Continuous Dynamical Systems - B, vol. 28, no. 2, pp. 1414–1435, 2023.
View at: Publisher Site | Google Scholar
L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, no. 3, pp. 338–353, 1965.
View at: Publisher Site | Google Scholar
K. B. Bao, L. B. Rong, and Q. M. Zhang, “Analysis of a stochastic SIRS model with interval parameters,” Discrete and Continuous Dynamical Systems - Series B, vol. 24, no. 9, pp. 4827–4849, 2019.
View at: Google Scholar
K. Bao, Q. Zhang, L. Rong, and X. N. Li, “Dynamics of an imprecise SIRS model with Lévy jumps,” Physica A: Statistical Mechanics and Its Applications, vol. 520, pp. 489–506, 2019.
View at: Publisher Site | Google Scholar
A. Das and M. Pal, “A mathematical study of an imprecise SIR epidemic model with treatment control,” Journal of Applied Mathematics and Computing, vol. 56, no. 1–2, pp. 477–500, 2018.
View at: Publisher Site | Google Scholar
D. Pal, G. Mahaptra, G. P. Samanta, and D. Pal, “Optimal harvesting of prey–predator system with interval biological parameters: a bioeconomic model,” Mathematical Biosciences, vol. 241, no. 2, pp. 181–187, 2013.
View at: Publisher Site | Google Scholar
D. Pal, G. S. Mahapatra, and G. P. Samanta, “Stability and bionomic analysis of fuzzy prey–predator harvesting model in presence of toxicity, a dynamic approach,” Bulletin of Mathematical Biology, vol. 78, no. 7, pp. 1493–1519, 2016.
View at: Publisher Site | Google Scholar
Q. L. Wang, Z. J. Liu, X. G. Zhang, and R. A. Cheke, “Incorporating prey refuge into a predator–prey system with imprecise parameter estimates,” Computational and Applied Mathematics, vol. 36, no. 2, pp. 1067–1084, 2015.
View at: Publisher Site | Google Scholar
S. Y. Chen, Z. J. Liu, L. W. Wang, and J. Hu, “Stability of a delayed competitive model with saturation effect and interval biological parameters,” Journal of Applied Mathematics and Computing, vol. 64, no. 1-2, pp. 1–15, 2020.
View at: Publisher Site | Google Scholar
X. Z. Zhang and H. Y. Zhao, “Bifurcation and optimal harvesting of a diffusive predator–prey system with delays and interval biological parameters,” Journal of Theoretical Biology, vol. 363, pp. 390–403, 2014.
View at: Publisher Site | Google Scholar
Y. Tang, D. Q. Huang, and W. N. Zhang, “Direct parametric analysis of an enzyme-catalyzed reaction model,” IMA Journal of Applied Mathematics, vol. 76, no. 6, pp. 876–898, 2011.
View at: Publisher Site | Google Scholar

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