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Can aphids be controlled by fungus? A mathematical model

Nicholas F. Britton

Nicholas F. Britton

Centre for Mathematical Biology and Department of Mathematical Sciences, University of BathBath, UK
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Britton, Nicholas F.
, 
Iulia Martina Bulai

Iulia Martina Bulai

Department of Information Engineering, University of PadovaPadova, Italy
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Bulai, Iulia Martina
, 
Stéphanie Saussure

Stéphanie Saussure

Norwegian Institute of Bioeconomy Research, Norwegian University of Life SciencesÅs, Norway
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Saussure, Stéphanie
, 
Niels Holst

Niels Holst

Department of Agroecology - Entomology and Plant Pathology, Århus UniversitySlagelse, Denmark
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Holst, Niels
 and 
Ezio Venturino

Ezio Venturino

Dipartimento di Matematica “Giuseppe Peano”, Università di TorinoTorino, Italy
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Venturino, Ezio
  
Jun 21, 2019
Can aphids be controlled by fungus? A mathematical model's Cover Image
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Published Online: Jun 21, 2019
Page range: 79 - 92
Received: Mar 05, 2019
Accepted: Apr 26, 2019
DOI: https://doi.org/10.2478/AMNS.2019.1.00009
Keywords
© 2019 Nicholas F. Britton, Iulia Martina Bulai, Stéphanie Saussure, Niels Holst, Ezio Venturino, published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 International License.

Fig. 1

Compartment diagram of the three-way (plant)-aphid-entomopathogen interactions.
Compartment diagram of the three-way (plant)-aphid-entomopathogen interactions.

Fig. 2

A two-dimensional cross section of the positively invariant set.
A two-dimensional cross section of the positively invariant set.

Fig. 3

A typical plot of the function Q(N) given in (8).
A typical plot of the function Q(N) given in (8).

Fig. 4

Sketch of the transcritical bifurcations undergone by the system’s equilibria.
Sketch of the transcritical bifurcations undergone by the system’s equilibria.

Fig. 5

One parameter bifurcation analysis of the populations N, E and F of the system (1) in terms of the parameter c, respectively. The continuous red curves represent the stable equilibrium points and the black one the unstable ones. The dotted green curve represent the maximum and minimum values of the oscillations triggered by the Hopf bifurcation. The other parameters values are β = 10, q = 1, γ = 5, b = 0.002, a = 2 and p = 1.2. The initial conditions are N = 1, E = 1 and F = 1.
One parameter bifurcation analysis of the populations N, E and F of the system (1) in terms of the parameter c, respectively. The continuous red curves represent the stable equilibrium points and the black one the unstable ones. The dotted green curve represent the maximum and minimum values of the oscillations triggered by the Hopf bifurcation. The other parameters values are β = 10, q = 1, γ = 5, b = 0.002, a = 2 and p = 1.2. The initial conditions are N = 1, E = 1 and F = 1.

Fig. 6

Zoomed portion of the Figure 5.
Zoomed portion of the Figure 5.

Fig. 7

Two parameter bifurcation analysis of the system (1) of the remaining six parameters as function of the bifurcation parameter c. Respectively, top to bottom and left to right, the former are b, β, s, γ, p, and q. The continuous blue line is the Hopf bifurcation curve. The dashed blue line represents the transcritical bifurcation from the coexistence equilibrium to disease-free steady state.
Two parameter bifurcation analysis of the system (1) of the remaining six parameters as function of the bifurcation parameter c. Respectively, top to bottom and left to right, the former are b, β, s, γ, p, and q. The continuous blue line is the Hopf bifurcation curve. The dashed blue line represents the transcritical bifurcation from the coexistence equilibrium to disease-free steady state.
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