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A Complete Mathematical Model For Trichoderma Fungi Kinetics

Asmae Hardoul, Zoubida Mghazli

TL;DR

This work develops a complete unstructured kinetic model for Trichoderma growth and cellulase production that integrates cellulose hydrolysis as a key flux, via four coupled ODEs for organic matter X, biomass B, substrate s, and product P with Monod growth and Leudeking-Piret production. It proves positivity of solutions and shows trajectories approach a continuum of non-hyperbolic equilibria E=(0,0,s^*,P^*) with X,B→0 and s^* in an admissible set, using Barbalat's lemma and invariant-manifold arguments. Numerical tests with literature data validate the model and illustrate how initial organic matter and mortality influence peak biomass and enzyme yield. The results provide a mathematically tractable framework for hydrolysis-limited fungal dynamics in the rhizosphere and motivate PDE extensions to capture spatial effects.

Abstract

We develop an unstructured mathematical model describing the growth kinetics of the Trichoderma fungus and the production of enzymes (cellulase) by degradation of a substrate (cellulose) in the rhizosphere. We integrate into this model the hydrolysis step of the organic matter and analyze the asymptotic behaviour of the obtained system. We show that our system evolves towards a global attractor consisting of infinite non-hyperbolic equilibria.

A Complete Mathematical Model For Trichoderma Fungi Kinetics

TL;DR

This work develops a complete unstructured kinetic model for Trichoderma growth and cellulase production that integrates cellulose hydrolysis as a key flux, via four coupled ODEs for organic matter X, biomass B, substrate s, and product P with Monod growth and Leudeking-Piret production. It proves positivity of solutions and shows trajectories approach a continuum of non-hyperbolic equilibria E=(0,0,s^*,P^*) with X,B→0 and s^* in an admissible set, using Barbalat's lemma and invariant-manifold arguments. Numerical tests with literature data validate the model and illustrate how initial organic matter and mortality influence peak biomass and enzyme yield. The results provide a mathematically tractable framework for hydrolysis-limited fungal dynamics in the rhizosphere and motivate PDE extensions to capture spatial effects.

Abstract

We develop an unstructured mathematical model describing the growth kinetics of the Trichoderma fungus and the production of enzymes (cellulase) by degradation of a substrate (cellulose) in the rhizosphere. We integrate into this model the hydrolysis step of the organic matter and analyze the asymptotic behaviour of the obtained system. We show that our system evolves towards a global attractor consisting of infinite non-hyperbolic equilibria.
Paper Structure (8 sections, 3 theorems, 50 equations, 10 figures, 2 tables)

This paper contains 8 sections, 3 theorems, 50 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Model and hypotheses
  3. Equilibrium points and asymptotic behaviour
  4. Positivity of the solution and equilibrium points
  5. Asymptotic behaviour
  6. Properties of the equilibrium points
  7. Numerical tests
  8. Conclusion

Key Result

Proposition 1

For any vector $(X_0,B_0,s_0,P_0)$ with non-negative components, the solution of the system model s1 with the initial condition $(X(0),B(0),s(0),P(0))=(X_0,B_0,s_0,P_0)$ exists and is unique and non-negative.

Figures (10)

  • Figure 1: Evolution of $X$, $B$, $s$ and $P$, using the parameters of Table \ref{['t1']}, Table \ref{['t2']} and the initial conditions $(X_0,B_0,s_0,P_0)=(45,15,50,0)$
  • Figure 2: Evolution of $X$, $B$, $s$, $P$, using the parameters of Table \ref{['t3']} and Table \ref{['t4']} with $(X_0,B_0,s_0,P_0)=(17,5,9.5,1.5)$
  • Figure 3: Evolution of $X$ for initial conditions $X^{i}_{0}$ for $i = 1,...,4$ .
  • Figure 4: Evolution of $B$ for initial conditions $X^{i}_{0}$ for $i = 1,...,4.$
  • Figure 5: Evolution of $P$ for initial conditions $X^{i}_{0}$ for $i = 1,...,4.$
  • ...and 5 more figures

Theorems & Definitions (6)

  • Proposition 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof