Table 1.
Parameter values per time unit (one day) used in the invasion analysis.
Table 2.
Parameter values per time unit (one day) used in the stability analysis.
Figure 1.
Bifurcation figures of the S-I-P-B dynamics, presenting maximum and minimum values (black circles), as well as equilibrium densities (red stars for when competitor (B) is not present, red circles when all the populations are present) of susceptible host (S), pathogen (P) and non-pathogenic (B) population densities in different combinations of outside-host growth rate of pathogen (rP) parameter values (rP = 0.001–0.5).
a) When susceptible host growth rate (rS) is high (rS = 1), increasing rP stabilizes the disease dynamics. b) Disease dynamics are cyclic when susceptible host growth rate (rS) is low (rS = 0.01). Used parameter values are shown in Table 2.
Figure 2.
Invasion analyses of a novel environmentally growing opportunist pathogen under outside-host competition situation in different combinations of the competition coefficient (fBP) parameter values and a) environmental transmission rate (β), b) release rate (Λ), c) virulence (α), d) pathogen mortality outside-host (μP), e) outside-host growth rate of pathogen (rP) and f) susceptible host growth rate (rS).
The parameter values used are shown in Table 1. The black area shows the parameter combinations for which the equilibrium dynamics are locally stable preventing the invasion of the pathogen. The white area shows where the dynamics become unstable enabling invasion of the new environmentally growing opportunist pathogen (P).
Figure 3.
Bifurcation figures of the S-I-P-B dynamics, presenting maximum and minimum values (black circles), as well as equilibrium densities when competitor (B) is not present (red stars) of susceptible host (S), pathogen (P) and non-pathogenic (B) population densities in different combinations of susceptible host growth rate of pathogen (rS) parameter values (rS = 0–1).
In all the figures fPB = 10−7. a) When outside-host growth rate of pathogen (rP) is 0.05 and the competition coefficient (fBP) is 10−5, decreasing rS destabilizes the disease dynamics. b) When rP is (0.05) and fBP is higher (10−4), pathogen population is able to increase and the dynamics are locally stable and all four populations coexist.
Figure 4.
Bifurcation figures of the S-I-P-B dynamics, presenting maximum and minimum values (black circles), as well as equilibrium densities when competitor (B) is not present (red stars) of susceptible host (S), pathogen (P) and non-pathogenic (B) population densities in different combinations of the competition coefficient (fBP) parameter values.
a) Dynamics are cyclic when susceptible host growth rate (rS) is 0.01, outside-host growth rate of pathogen (rP) is 0.05 and fBP varies between 0 and 10−5. b) When rS = 0.01, rP = 0.05 and fBP = 10−7–10−4, dynamics are first cyclic and the densities of P, S and I are decreasing as fBP and the density of B increase. As fBP increases further, pathogen population stabilizes close to zero and the density of S starts to increase. c) When rS = 1, rP = 0.5, fPB = 10−7 and fBP = 10−8–10−4, the coexistence dynamics are locally stable. As fBP increases, pathogen goes extinct. The parameter values used are shown in Table 2.