Table 1 Parameter values used in the analyses

α

Virulence (Mortality of the infected hosts due to infection)

0.1 (day⁻¹)

in all the analysis

β

Pathogen transmission rate to susceptible hosts from environment

10⁻⁵ (day⁻¹)

in all the analysis

Λ

Pathogen release rate from infected hosts when they die

10⁹

in all the analysis

β _B

Phage-resistant pathogen transmission rate to susceptible hosts from environment

0–10⁻¹⁰ (day^− 1)

in Fig. 3

0 (day^− 1)

in Figs. 1, 2; Additional file 1: Figure S1; Additional file 3: Figure S2 and Additional file 4: Figure S3

Λ _B

Phage-resistant pathogen release rate from infected hosts when they die

10⁵

in Fig. 3

0

in Figs. 1, 2; Additional file 1: Figure S1; Additional file 3: Figure S2 and Additional file 4: Figure S3

β _F

Phage transmission rate to pathogens from the environment

10⁻⁸–10^− 4 (day^− 1)

in Fig. 1

10^− 7 (day^− 1)

in Fig. 2

10^− 3 (day^− 1)

in Fig. 3 and Additional file 4: Figure S3

0

in Additional file 1: Figure S1

10^− 9–10^− 5 (day^− 1)

in Additional file 3: Figure S2

Λ _F

Phage burst size from infected pathogen

100

in Figs. 1, 2 and 3 and Additional file 4: Figure S3

0

in Additional file 1: Figure S1

0.0001–10

in Additional file 3: Figure S2

r _S

Susceptible host growth rate

0.1 (day^− 1)

in all the analysis

r _P

Pathogen growth rate outside-host

3 (day⁻¹)

in all the analysis

r _B

Phage-resistant bacteria growth rate outside-host

3 (day⁻¹)

in Fig. 3

0 (day^− 1)

in Figs. 1, 2; Additional file 1: Figure S1; Additional file 3: Figure S2 and Additional file 4: Figure S3

K

Constant parameter K modifying the strength of density-dependence of the environmental growth rate

10⁹

in all the analysis

μ _SI

Mortality of the susceptible and infected hosts due to other reasons than infection

0.001 (day⁻¹)

in all the analysis

μ _P

Pathogen mortality outside-host

0.1 (day⁻¹)

in all the analysis

μ _B

Phage-resistant bacteria mortality outside-host

0.1 (day⁻¹)

in Fig. 3

0 (day^− 1)

in Figs. 1, 2; Additional file 1: Figure S1; Additional file 3: Figure S2 and Additional file 4: Figure S3

μ _F

Phage decay rate outside-host

0.1 (day^− 1)

in Figs. 1, 2, 3; Additional file 3: Figure S2 and Additional file 4: Figure S3

0 (day^− 1)

in Additional file 1: Figure S1

δ

Recovery of the hosts from infection

0 (day^− 1)

Figs. 1, 3; Additional file 1: Figure S1; Additional file 3: Figure S2 and Additional file 4: Figure S3

0–0.8 (day^− 1)

in Fig. 2