Stability Analysis of Fixed Points in Forest Biomass Depletion Model

Vivi Ramdhani; Yolanda Rahmi Safitri

doi:10.31958/js.v15i2.9240

Stability Analysis of Fixed Points in Forest Biomass Depletion Model

Vivi Ramdhani, Yolanda Rahmi Safitri

Abstract

The main focus of this article is to present a model that explains the depletion in forest biomass due to population size , population pressure , and industrialization . This research explores the complex relationship between forest biomass, population growth, and industrialization. The model generates four fixed points that are all non-negative, and they are known as and . The article proceeds to analyze the stability of these fixed points in the context of forest biomass depletion. It is discovered that the fixed points and are saddle points and not stable, while the fixed point is stable if it meets certain conditions. The article concludes by carrying out numerical simulations to determine the equilibrium point of the model, which shows that forest resource biomass stability declines as population size, population pressure, and industrialization increase. The simulations reveal that population growth results in a depletion in forest resource biomass, while the opposite is true for the positive impact that forest resource biomass on population levels. Consequently, it is essential to regulate population density and industrialization to manage population growth and protect forest resources.

Keywords

Fixed points; Forest biomass; Population; Stability.

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References

Anggriani, I., Nurhayati, S., & Subchan, S. (2018). Analisis Kestabilan Model Penurunan Sumber Daya Hutan Akibat Industri. Limits: Journal of Mathematics and Its Applications, 15(1), 31. https://doi.org/10.12962/limits.v15i1.3560

Boyce, W. E., & DiPrima, R. E. (2012). Elementary Differential Equation and Boundary Value Problems.

Dubey, B., Sharma, S., Sinha, P., & Shukla, J. B. (2009). Modelling the depletion of forestry resources by population and population pressure augmented industrialization. Applied Mathematical Modelling, 33(7), 3002–3014. https://doi.org/10.1016/j.apm.2008.10.028

Dubey, B., Upadhyay, R. K., & Hussain, J. (2002). Effects of industrialization and pollution on resource biomass: A mathematical model. In Ecological Modelling (Issues 1–2). https://doi.org/10.1016/S0304-3800(03)00168-6

Khalil, H. K. (2002). Nonlinear Systems. In Upper Saddle River. NJ: Prentice Hall.

Mohamad, R., Rauf, M. D. A., & Lakisa, N. (2019). Model Matematika Kerusakan Hutan dengan Memperhatikan Faktor Industri dan Kebakaran. EULER: Jurnal Matematika, Sains Dan Teknologi, 7(1), 6–14.

Ramdhani, V., Jaharuddin, & Nugrahani, E. H. (2015). Dynamical system of modelling the depletion of forestry resources due to crowding by industrialization. Applied Mathematical Sciences, 9(81–84), 4067–4079. https://doi.org/10.12988/ams.2015.53259

Robinson, J. (2004). An Introduction to Ordinary Differential Equations.UK: Cambridge University Press.

Shukla, J. B., Kusum, L., & Misra, A. K. (2011). Resource By Population and Industrialization : Effect of Technology on Its Conservation. Natural Resource Modeling, 24(2), 242–267.

Sundar, S., Swaroop, N., & Naresh, R. (2017). Modeling the Effect of Population and Population Augmented Industrialization on Forestry Resources. European Journal of Engineering Research and Science, 2(1), 65. https://doi.org/10.24018/ejers.2017.2.1.247

Wakhidah, U. U. S., Nugraheni, K., & Winarni, W. (2022). Analysis of Mangrove Forest Resource Depletion Models due to The Opening of Fish Pond Land with Time Delay. Jurnal ILMU DASAR, 23(1), 65. https://doi.org/10.19184/jid.v23i1.23889

DOI: http://dx.doi.org/10.31958/js.v15i2.9240