Modeling the effects of rising carbon dioxide levels in atmosphere on urban life and forest resources

Abstract

A nonlinear mathematical model is proposed and analyzed to investigate the effects of rising atmospheric carbon dioxide and global warming on urban human population and forest resources. The continuous growing urban human population along with urban development puts many negative consequences in the environment, aggravate the global warming and climate change simultaneously. In the proposed model, it is assumed that the only inherent growth of forest resources is affected by the increasing urban human population and global warming. However, increasing rampant urban development damage the carrying capacity of forest resources abruptly which is harmful for all ecological system. Six dynamical variables like forest resources, urban human population, population pressure, urbanisation, carbon dioxide concentration and global warming are incorporated as a nonlinear dynamical system of differential equations. The proposed model is analyzed by the both of way as qualitative and quantitative analysis. The region of attraction, finding of equilibriums with their existence, stabilities and uniform persistence are obtained by using the qualitative analysis of the proposed model. Likewise, for getting numerical results of the model is analyzed quantitatively. It is observed that global warming affect the health of urban human population as well as forest resources in the long run.

Appendices

Appendix A

In the model system (1), from the first equation of the model

$$\frac{dF}{{dt}} \le rF(1 - \frac{F}{L}) + \gamma \gamma_{1} CF$$

$$F \le \frac{L}{r}(r + \gamma \gamma_{1} C) = F_{m}$$

This provides $0 \le F \le F_{m}$.

From the second equation of the model

$$\frac{dN}{{dt}} \le sN(1 - \frac{N}{k}) + \beta_{1} \beta_{2} NF$$

$$N \le \frac{k}{s}(s + \beta_{1} \beta_{2} F_{m} ) = N_{m}$$

This provides $0 \le N \le N_{m}$.

From the third equation of the model

$$\frac{dP}{{dt}} \le \eta N - \lambda_{0} P$$

$$P \le \frac{\eta }{{\lambda_{0} }}N_{m} = P_{m}$$

This provides $0 \le P \le P_{m}$.

From the fourth equation of the model

$$\frac{dU}{{dt}} \le Q + k_{1} P - \theta_{0} U$$

$$U \le \frac{{Q + k_{1} P_{m} }}{{\theta_{0} }} = U_{m}$$

This provides $0 \le U \le U_{m}$.

From the fifth equation of the model

$$\frac{dC}{{dt}} \le A + \theta N - \delta C + \theta_{1} U$$

$$C \le \frac{{A + \theta N_{m} + \theta_{1} U_{m} }}{\delta } = C_{m}$$

This provides $0 \le C \le C_{m}$.

From the sixth equation of the model

$$\frac{dG}{{dt}} \le B + \lambda C + \lambda_{1} N - \varphi G$$

$$G \le \frac{{B + \lambda C_{m} + \lambda_{1} N_{m} }}{{\varphi + \lambda_{2} F}} = G_{m}$$

This provides $0 \le G \le G_{m}$.

Appendix B

Theorem 3.1

The model is locally asymptotically stable around the equilibrium point if the following conditions hold.

• (i) $4(\mu \lambda_{1} \beta_{2} + \alpha_{2} \lambda_{2} G^{ * } )^{2} < \lambda_{1} \alpha_{2} \beta_{2} (\frac{r}{L} + \alpha_{1} U^{ * } )(\varphi + \lambda_{2} F^{ * } )$

• (ii) $6\beta_{2} C^{ * } k\left( {\frac{{\theta \gamma_{1} }}{{C^{ * } }} - \frac{{\alpha_{0} }}{{\beta_{2} }}} \right)^{2} < s\gamma_{1} (\delta + \gamma F^{ * } )$

• (iii) $4k\eta^{2} \beta_{2} < s\lambda_{0}$

• (iv) $6C^{ * } \alpha_{2} \lambda^{2} < \lambda_{1} \beta_{2} \gamma_{1} (\delta + \gamma F^{ * } )(\varphi + \lambda_{2} F^{ * } )$

• (v) $\max \left\{ {\frac{{6( - \alpha_{1} F^{ * } )^{2} }}{{\left( {\frac{r}{L} + \alpha_{1} U^{ * } } \right)\theta_{0} }},\frac{{9\gamma_{1} \theta_{1}^{2} }}{{C^{ * } \theta_{0} (\delta + \gamma F^{ * } )}}} \right\} < \frac{{\lambda_{0} \theta_{0} }}{{6(k_{1} )^{2} }}$

Proof: Following are the transformations that can be made:

$F = F^{ * } + x_{1}$,$N = N^{ * } + x_{2}$,$P = P^{ * } + x_{3}$,$U = U^{ * } + x_{4}$,$C = C^{ * } + x_{5}$,$G = G^{ * } + x_{6}$

$$V = \frac{1}{2}[\frac{{x_{1}^{2} }}{{F^{ * } }} + l_{1} \frac{{x_{2}^{2} }}{{N^{ * } }} + l_{2} x_{3}^{2} + l_{3} x_{4}^{2} + l_{4} x_{5}^{2} + l_{5} x_{6}^{2} ]$$

(47)

Using the derivative $V$ of the equation with regard to $t$, we acquire

$$\frac{dV}{{dt}} = \frac{{x_{1} }}{{F^{ * } }}\frac{{dx_{1} }}{dt} + l_{1} \frac{{x_{2} }}{{N^{ * } }}\frac{{dx_{2} }}{dt} + l_{2} x_{3} \frac{{dx_{3} }}{dt} + l_{3} x_{4} \frac{{dx_{4} }}{dt} + l_{4} x_{5} \frac{{dx_{5} }}{dt} + l_{5} x_{6} \frac{{dx_{6} }}{dt}$$

(48)

$$\begin{gathered} \frac{{dV}}{{dt}} = - \left( {\frac{r}{L} + \alpha _{1} U^{*} } \right)x_{1}^{2} - l_{1} \frac{s}{k}x_{2}^{2} - l_{2} \lambda _{0} x_{3}^{2} - l_{3} \theta _{0} x_{4}^{2} - l_{4} (\delta + \gamma F^{*} )x_{5}^{2} - l_{5} (\varphi + \lambda _{2} F^{*} )x_{6}^{2} \hfill \\ + ( - \beta _{1} + l_{1} \beta _{1} \beta _{2} )x_{1} x_{2} - \alpha _{1} F^{*} x_{1} x_{4} + (\gamma \gamma _{1} - l_{4} \gamma C^{*} )x_{1} x_{5} + ( - \mu - l_{5} \lambda _{2} G^{*} )x_{1} x_{6} \hfill \\ + (l_{4} \theta - l_{1} \alpha _{0} )x_{2} x_{5} + l_{2} \eta x_{2} x_{3} + k_{1} l_{3} x_{3} x_{4} + l_{4} \theta _{1} x_{4} x_{5} + l_{5} \lambda x_{5} x_{6} + ( - l_{1} \alpha _{2} + l_{5} \lambda _{1} )x_{2} x_{6} \hfill \\ \end{gathered}$$

(49)

If we choose

$( - \beta_{1} + l_{1} \beta_{1} \beta_{2} ) = 0,$ this gives $l_{1} = \frac{1}{{\beta_{2} }}$.

$(\gamma \gamma_{1} - l_{4} \gamma C^{ * } ) = 0$, this gives $l_{4} = \frac{{\gamma_{1} }}{{C^{ * } }}$.

$(l_{5} \lambda_{1} - l_{1} \alpha_{2} ) = 0$, this gives $l_{5} = \frac{{\alpha_{2} }}{{\lambda_{1} \beta_{2} }}$.

$$\begin{gathered} \frac{{dV}}{{dt}} = - \frac{1}{2}(\frac{r}{L} + \alpha _{1} U^{*} )x_{1}^{2} - \alpha _{1} F^{*} x_{1} x_{4} - \frac{1}{3}l_{3} \theta _{0} x_{4}^{2} \hfill \\ - \frac{1}{2}(\frac{r}{L} + \alpha _{1} U^{*} )x_{1}^{2} - (\mu + l_{5} \lambda _{2} G^{*} )x_{1} x_{6} - \frac{1}{2}l_{5} (\varphi + \lambda _{2} F^{*} )x_{6}^{2} \hfill \\ - \frac{1}{2}l_{1} \frac{s}{k}x_{2}^{2} + (l_{4} \theta - l_{1} \alpha _{0} )x_{2} x_{5} - \frac{1}{3}l_{4} (\delta + \gamma F^{*} )x_{5}^{2} \hfill \\ - \frac{1}{2}l_{1} \frac{s}{k}x_{2}^{2} + l_{2} \eta x_{2} x_{3} - \frac{1}{2}l_{2} \lambda _{0} x_{3}^{2} \hfill \\ - \frac{1}{2}l_{2} \lambda _{0} x_{3}^{2} + k_{1} l_{3} x_{3} x_{4} - \frac{1}{3}l_{3} \theta _{0} x_{4}^{2} \hfill \\ - \frac{1}{3}l_{3} \theta _{0} x_{4}^{2} + l_{4} \theta _{1} x_{4} x_{5} - \frac{1}{3}l_{4} (\delta + \gamma F^{*} )x_{5}^{2} \hfill \\ - \frac{1}{3}l_{4} (\delta + \gamma F^{*} )x_{5}^{2} + l_{5} \lambda x_{5} x_{6} - \frac{1}{2}l_{5} (\varphi + \lambda _{2} F^{*} )x_{6}^{2} \hfill \\ \end{gathered}$$

(50)

If we choose $l_{2} = 1$, we acquire.

Now, $\frac{dV}{{dt}}$ will be negative if.

• (i) $( - \alpha_{1} F^{ * } )^{2} < \frac{{l_{3} }}{6}(\frac{r}{L} + \alpha_{1} U^{ * } )\theta_{0}$

which can be written as

$$\frac{{6( - \alpha_{1} F^{ * } )^{2} }}{{\left( {\frac{r}{L} + \alpha_{1} U^{ * } } \right)\theta_{0} }} < l_{3}$$

(51)

• (ii) $(\mu + l_{5} \lambda_{2} G^{ * } )^{2} < \frac{1}{4}l_{5} (\frac{r}{L} + \alpha_{1} U^{ * } )(\varphi + \lambda_{2} F^{ * } )$

.

which can be written as

$$4(\mu \lambda_{1} \beta_{2} + \alpha_{2} \lambda_{2} G^{ * } )^{2} < \lambda_{1} \alpha_{2} \beta_{2} (\frac{r}{L} + \alpha_{1} U^{ * } )(\varphi + \lambda_{2} F^{ * } )$$

(52)

• (iii) $( - l_{1} \alpha_{0} + l_{4} \theta )^{2} < \frac{1}{6}l_{1} l_{4} \frac{s}{k}(\delta + \gamma F^{ * } )$

which can be written as

$$6\beta_{2} C^{ * } k\left( {\frac{{\theta \gamma_{1} }}{{C^{ * } }} - \frac{{\alpha_{0} }}{{\beta_{2} }}} \right)^{2} < s\gamma_{1} (\delta + \gamma F^{ * } )$$

(53)

• (iv) $(l{}_{2}\eta )^{2} < \frac{1}{4}l_{1} l_{2} \frac{s}{k}\lambda_{0}$

which can be written as

$$4k\eta^{2} \beta_{2} < s\lambda_{0}$$

(54)

• (xxii) $(k_{1} l_{3} )^{2} < \frac{1}{6}l_{2} l_{3} \lambda_{0} \theta_{0}$

which can be written as

$$l{}_{3} < \frac{{\lambda_{0} \theta_{0} }}{{6(k_{1} )^{2} }}$$

(55)

• (vi) $(l_{4} \theta_{1} )^{2} < \frac{1}{9}l_{3} l_{4} \theta_{0} (\delta + \gamma F^{ * } )$

which can be written as

$$\frac{{9\gamma_{1} \theta_{1}^{2} }}{{C^{ * } \theta_{0} (\delta + \gamma F^{ * } )}} < l_{3}$$

(56)

• (vii) $(l_{5} \lambda )^{2} < \frac{1}{6}l_{4} l_{5} (\varphi + \lambda_{2} F^{ * } )(\delta + \gamma F^{ * } )$

which can be written as

$$6C^{ * } \alpha_{2} \lambda^{2} < \lambda_{1} \beta_{2} \gamma_{1} (\delta + \gamma F^{ * } )(\varphi + \lambda_{2} F^{ * } )$$

(57)

From Equations (51), (55) and (56), we acquire

$$\max \left\{ {\frac{{6( - \alpha_{1} F^{ * } )^{2} }}{{\left( {\frac{r}{L} + \alpha_{1} U^{ * } } \right)\theta_{0} }},\frac{{9\gamma_{1} \theta_{1}^{2} }}{{C^{ * } \theta_{0} (\delta + \gamma F^{ * } )}}} \right\} < l_{3} < \frac{{\lambda_{0} \theta_{0} }}{{6(k_{1} )^{2} }}$$

(58)

which ultimately reduces to

$$\max \left\{ {\frac{{6( - \alpha_{1} F^{ * } )^{2} }}{{\left( {\frac{r}{L} + \alpha_{1} U^{ * } } \right)\theta_{0} }},\frac{{9\gamma_{1} \theta_{1}^{2} }}{{C^{ * } \theta_{0} (\delta + \gamma F^{ * } )}}} \right\} < \frac{{\lambda_{0} \theta_{0} }}{{6(k_{1} )^{2} }}$$

(59)

Appendix C

Theorem 3.2

The model is globally asymptotically stable around the equilibrium point if the following conditions hold.

• (i). $4(\mu \lambda_{1} \beta_{2} + \alpha_{2} \lambda_{2} G^{ * } )^{2} < \lambda_{1} \alpha_{2} \beta_{2} (\frac{r}{L} + \alpha_{1} U^{ * } )(\varphi + \lambda_{2} F^{ * } )$

• (ii). $6\beta_{2} C^{ * } k\left( {\frac{{\theta \gamma_{1} }}{{C^{ * } }} - \frac{{\alpha_{0} }}{{\beta_{2} }}} \right)^{2} < s\gamma_{1} (\delta + \gamma F^{ * } )$

• (iii). $4k\eta^{2} \beta_{2} < s\lambda_{0}$

• (iv). $6C^{ * } \alpha_{2} \lambda^{2} < \lambda_{1} \beta_{2} \gamma_{1} (\delta + \gamma F^{ * } )(\varphi + \lambda_{2} F^{ * } )$

  $$\max \left\{ {\frac{{6( - \alpha_{1} F^{ * } )^{2} }}{{\left( {\frac{r}{L} + \alpha_{1} U^{ * } } \right)\theta_{0} }},\frac{{9\gamma_{1} \theta_{1}^{2} }}{{C^{ * } \theta_{0} (\delta + \gamma F^{ * } )}}} \right\} < \frac{{\lambda_{0} \theta_{0} }}{{6(k_{1} )^{2} }}$$

Proof: Following are the transformations that can be made:

$$U = (F - F^{*} - F^{*} \ln \frac{F}{{F^{*} }}) + m_{1} \left( {N - N^{*} - N^{*} \ln \frac{N}{{N^{*} }}} \right) + \frac{1}{2}m_{2} (P - P^{*} ) + \frac{1}{2}m_{3} (U - U^{*} ) + \frac{1}{2}m_{4} (C - C^{*} ) + \frac{1}{2}m_{5} (G - G^{*} )$$

(60)

where $m_{1} ,\,m_{2} ,\,m_{3} ,\,m_{4} \,and\,m_{5}$ are positive constants.

Using the derivative $U$ of the equation with regard to $t$, we acquire

$$\begin{gathered} \frac{{dU}}{{dt}} = \frac{{(F - F^{*} )}}{F}\frac{{dF}}{{dt}} + m_{1} \frac{{(N - N^{*} )}}{N}\frac{{dN}}{{dt}} + m_{2} (P - P^{*} )\frac{{dP}}{{dt}} + m_{3} (U - U^{*} )\frac{{dU}}{{dt}} \hfill \\ + m_{4} (C - C^{*} )\frac{{dC}}{{dt}} + m_{5} (G - G^{*} )\frac{{dG}}{{dt}} \hfill \\ \end{gathered}$$

(61)

$$\begin{gathered} \frac{{dU}}{{dt}} = - \left( {\frac{r}{L} + \alpha _{1} U} \right)(F - F^{*} )^{2} - m_{1} \frac{s}{k}(N - N^{*} )^{2} - m_{2} \lambda _{0} (P - P^{*} )^{2} - m_{3} \theta _{0} (U - U^{*} )^{2} \hfill \\ - m_{4} (\delta + \gamma F)(C - C^{*} )^{2} - m_{5} (\varphi + \lambda _{2} F)(G - G^{*} )^{2} \hfill \\ + ( - \beta _{1} + m_{1} \beta _{1} \beta _{2} )(F - F^{*} )(N - N^{*} ) - \alpha _{1} F^{*} (F - F^{*} )(U - U^{*} ) \hfill \\ + (\gamma \gamma _{1} - m_{4} \gamma C)(F - F^{*} )(C - C^{*} ) + (m_{4} \theta - m_{1} \alpha _{0} )(N - N^{*} )(C - C^{*} ) \hfill \\ + m_{2} \eta (N - N^{*} )(P - P^{*} ) + k_{1} m_{3} (U - U^{*} )(P - P^{*} ) + m_{4} \theta _{1} (U - U^{*} )(C - C^{*} ) \hfill \\ + m_{5} \lambda (C - C^{*} )(G - G^{*} ) + (m_{5} \lambda _{1} - m_{1} \alpha _{2} )(N - N^{*} )(G - G^{*} ) - (\mu + m_{5} \lambda _{2} G)(F - F^{*} )(G - G^{*} ) \hfill \\ \end{gathered}$$

(62)

If we choose.

$( - \beta_{1} + m_{1} \beta_{1} \beta_{2} ) = 0,$ this gives $m_{1} = \frac{1}{{\beta_{2} }}$.

$(\gamma \gamma_{1} - m_{4} \gamma C) = 0$, this gives $m_{4} = \frac{{\gamma_{1} }}{C}$.

$(m_{5} \lambda_{1} - m_{1} \alpha_{2} ) = 0$, this gives $m_{5} = \frac{{\alpha_{2} }}{{\lambda_{1} \beta_{2} }}$

$$\begin{gathered} \frac{{dU}}{{dt}} = - \frac{1}{2}(\frac{r}{L} + \alpha _{1} U)(F - F^{*} )^{2} - \alpha _{1} F^{*} (F - F^{*} )(U - U^{*} ) - \frac{1}{3}m_{3} \theta _{0} (U - U^{*} )^{2} \hfill \\ - \frac{1}{2}(\frac{r}{L} + \alpha _{1} U)(F - F^{*} )^{2} - (\mu + m_{5} \lambda _{2} G)(F - F^{*} )(G - G^{*} ) - \frac{1}{2}m_{5} (\varphi + \lambda _{2} F^{*} )(G - G^{*} )^{2} \hfill \\ - \frac{1}{2}m_{1} \frac{s}{k}(N - N^{*} )^{2} + (m_{4} \theta - m_{1} \alpha _{0} )(N - N^{*} )(C - C^{*} ) - \frac{1}{3}m_{4} (\delta + \gamma F^{*} )(C - C^{*} )^{2} \hfill \\ - \frac{1}{2}m_{1} \frac{s}{k}(N - N^{*} )^{2} + m_{2} \eta (N - N^{*} )(P - P^{*} ) - \frac{1}{2}m_{2} \lambda _{0} (P - P^{*} )^{2} \hfill \\ - \frac{1}{2}m_{2} \lambda _{0} (P - P^{*} )^{2} + k_{1} m_{3} (P - P^{*} )(U - U^{*} ) - \frac{1}{3}m_{3} \theta _{0} (U - U^{*} )^{2} \hfill \\ - \frac{1}{3}m_{3} \theta _{0} (U - U^{*} )^{2} + m_{4} \theta _{1} (U - U^{*} )(C - C^{*} ) - \frac{1}{3}m_{4} (\delta + \gamma F^{*} )(C - C^{*} )^{2} \hfill \\ - \frac{1}{3}m_{4} (\delta + \gamma F)(C - C^{*} )^{2} + m_{5} \lambda (C - C^{*} )(G - G^{*} ) - \frac{1}{2}m_{5} (\varphi + \lambda _{2} F^{*} )(G - G^{*} )^{2} \hfill \\ \end{gathered}$$

(63)

If we choose $m_{2} = 1$, we acquire.

Now, $\frac{dU}{{dt}}$ will be negative if.

• (i) $( - \alpha_{1} F^{ * } )^{2} < \frac{{m_{3} }}{6}\left( {\frac{r}{L} + \alpha_{1} U} \right)\theta_{0}$

which can be written as

$$\frac{{6( - \alpha_{1} F^{ * } )^{2} }}{{\left( {\frac{r}{L} + \alpha_{1} U} \right)\theta_{0} }} < m_{3}$$

(64)

• (ii) $(\mu + m_{5} \lambda_{2} G^{ * } )^{2} < \frac{1}{4}m_{5} (\frac{r}{L} + \alpha_{1} U^{ * } )(\varphi + \lambda_{2} F^{ * } )$

which can be written as

$$4(\mu \lambda_{1} \beta_{2} + \alpha_{2} \lambda_{2} G^{ * } )^{2} < \lambda_{1} \alpha_{2} \beta_{2} (\frac{r}{L} + \alpha_{1} U^{ * } )(\varphi + \lambda_{2} F^{ * } )$$

(65)

• (iii) $( - m_{1} \alpha_{0} + m_{4} \theta )^{2} < \frac{1}{6}m_{1} m_{4} \frac{s}{k}(\delta + \gamma F^{ * } )$

.

which can be written as

$$6\beta_{2} C^{ * } k\left( {\frac{{\theta \gamma_{1} }}{{C^{ * } }} - \frac{{\alpha_{0} }}{{\beta_{2} }}} \right)^{2} < s\gamma_{1} (\delta + \gamma F^{ * } )$$

(66)

• (iv) $(m{}_{2}\eta )^{2} < \frac{1}{4}m_{1} m_{2} \frac{s}{k}\lambda_{0}$

$$4k\eta^{2} \beta_{2} < s\lambda_{0}$$

(67)

• (xxii) $(k_{1} m_{3} )^{2} < \frac{1}{6}m_{2} m_{3} \lambda_{0} \theta_{0}$

which can be written as

$$m{}_{3} < \frac{{\lambda_{0} \theta_{0} }}{{6(k_{1} )^{2} }}$$

(68)

• (vi) $(m_{4} \theta_{1} )^{2} < \frac{1}{9}m_{3} m_{4} \theta_{0} (\delta + \gamma F^{ * } )$

which can be written as

$$\frac{{9\gamma_{1} \theta_{1}^{2} }}{{C^{ * } \theta_{0} (\delta + \gamma F^{ * } )}} < m_{3}$$

(69)

• (vii) $(m_{5} \lambda )^{2} < \frac{1}{6}m_{4} m_{5} (\varphi + \lambda_{2} F^{ * } )(\delta + \gamma F^{ * } )$

$$6C^{ * } \alpha_{2} \lambda^{2} < \lambda_{1} \beta_{2} \gamma_{1} (\delta + \gamma F^{ * } )(\varphi + \lambda_{2} F^{ * } )$$

(70)

From Eqs. (64), (68) and (69), we acquire

$$\max \left\{ {\frac{{6( - \alpha_{1} F^{ * } )^{2} }}{{\left( {\frac{r}{L} + \alpha_{1} U^{ * } } \right)\theta_{0} }},\frac{{9\gamma_{1} \theta_{1}^{2} }}{{C^{ * } \theta_{0} (\delta + \gamma F^{ * } )}}} \right\} < m_{3} < \frac{{\lambda_{0} \theta_{0} }}{{6(k_{1} )^{2} }}$$

(71)

which ultimately reduces to

$$\max \left\{ {\frac{{3( - \alpha_{1} F^{ * } )^{2} }}{{\left( {\frac{r}{L} + \alpha_{1} U^{ * } } \right)\theta_{0} }},\frac{{9\gamma_{1} \theta_{1}^{2} }}{{C^{ * } \theta_{0} (\delta + \gamma F^{ * } )}}} \right\} < \frac{{\lambda_{0} \theta_{0} }}{{6(k_{1} )^{2} }}$$

(72)

Appendix D

Theorem 3.3

The model system (1) exhibits uniform persistence, if the following conditions hold true:

• (i). $r + \gamma \gamma_{1} C_{m} > \beta_{1} N_{m} + \mu G_{m}$

• (ii). $s > \alpha_{0} C_{m} + \alpha_{2} G_{m}$

under the region of attraction $\Omega$ of Lemma 3.1.

Proof: From the first equation of the model (1), we have

$$\begin{gathered} \frac{{dF}}{{dt}} \ge rF - \frac{r}{L}F^{2} - \beta _{1} N_{m} F - \alpha _{1} F^{2} U_{m} + \gamma \gamma _{1} C_{m} F - \mu G_{m} F \hfill \\ \quad \, = (r - \beta _{1} N_{m} + \gamma \gamma _{1} C_{m} - \mu G_{m} )F - (\frac{r}{L} + \alpha _{1} U_{m} )F^{2} \hfill \\ \end{gathered}$$

This further implies that, provided

$$\mathop {\lim }\limits_{t \to \infty } \,\inf \,F(t) \ge \frac{{L(r - \beta_{1} N_{m} + \gamma \gamma_{1} C_{m} - \mu G_{m} )}}{{(r + L\alpha_{1} U_{m} )}} = F_{M} (say)$$

$$r + \gamma \gamma_{1} C_{m} > \beta_{1} N_{m} + \mu G_{m}$$

(73)

From the second equation of the model (1), we have

$$\begin{gathered} \frac{{dN}}{{dt}} \ge sN - \frac{s}{k}N^{2} - \alpha _{0} NC_{m} - \alpha _{2} G_{m} N \hfill \\ \quad \quad = (s - \alpha _{0} C_{m} - \alpha _{2} G_{m} )N - \frac{s}{k}N^{2} \hfill \\ \end{gathered}$$

This further implies that, provided

$$\mathop {\lim }\limits_{t \to \infty } \,\inf \,N(t) \ge \frac{{k(s - \alpha_{0} C_{m} - \alpha_{2} G_{m} )}}{s} = N_{M}\, (\text{say})$$

$$s > \alpha_{0} C_{m} + \alpha_{2} G_{m}$$

(74)

From the third equation of the model (1), we have

$$\frac{dP}{{dt}} \ge \eta N_{M} - \lambda_{0} P$$

This further implies that

$$\mathop {\lim }\limits_{t \to \infty } \,\inf \,P(t) \ge \frac{\eta }{{\lambda_{0} }}N_{M} = P_{M}\, (\text{say})$$

From the fourth equation of the model (1), we have

$$\frac{dU}{{dt}} \ge Q + k{}_{1}P_{M} - \theta_{0} U$$

This further implies that

$$\mathop {\lim }\limits_{t \to \infty } \,\inf \,U(t) \ge \frac{{Q + k_{1} P{}_{M}}}{{\theta_{0} }} = U_{M}$$

From the fifth equation of the model (1), we have

$$\frac{dC}{{dt}} \ge A + \theta N_{M} - \delta C - \gamma CF_{M} + \theta_{1} U_{M}$$

This further implies that

$$\mathop {\lim }\limits_{t \to \infty } \,\inf \,C(t) \ge \frac{{A + \theta N_{M} + \theta_{1} U_{M} }}{{\delta + \gamma F_{M} }} = C_{M} (say)$$

From the sixth equation of the model (1), we have

$$\frac{dG}{{dt}} \ge B + \lambda C_{M} + \lambda_{1} N_{M} - \varphi G$$

This further implies that

$$\mathop {\lim }\limits_{t \to \infty } \,\inf \,G(t) \ge \frac{{B + \lambda C_{M} + \lambda_{1} N_{M} }}{\varphi } = G_{M} (say)$$

The supremum values of the dynamical variables obtained from Lemma 3.1 combined with their aforementioned infimum values can be written as.

• (i). $F_{M} \le \mathop {\lim }\limits_{t \to \infty } \,\inf F(t) \le \mathop {\lim }\limits_{t \to \infty } \sup F(t) \le F_{m} ,$

• (ii). $N_{M} \le \mathop {\lim }\limits_{t \to \infty } \,\inf N(t) \le \mathop {\lim }\limits_{t \to \infty } \sup N(t) \le N_{m} ,$

• (iii). $P_{M} \le \mathop {\lim }\limits_{t \to \infty } \,\inf P(t) \le \mathop {\lim }\limits_{t \to \infty } \sup P(t) \le P_{m} ,$

• (iv). $U_{M} \le \mathop {\lim }\limits_{t \to \infty } \,\inf U(t) \le \mathop {\lim }\limits_{t \to \infty } \sup U(t) \le U_{m} ,$

• (v). $C_{M} \le \mathop {\lim }\limits_{t \to \infty } \,\inf C(t) \le \mathop {\lim }\limits_{t \to \infty } \sup C(t) \le C_{m} ,$

• (vi). $G_{M} \le \mathop {\lim }\limits_{t \to \infty } \,\inf G(t) \le \mathop {\lim }\limits_{t \to \infty } \sup G(t) \le G_{m} ,$

According to the aforementioned arguments, the model system (1) displays uniform persistence in the following circumstances:

$$r + \gamma \gamma_{1} C_{m} > \beta_{1} N_{m} + \mu G_{m}$$

and $s > \alpha_{0} C_{m} + \alpha_{2} G_{m}s > \alpha_{0} C_{m} + \alpha_{2} G_{m}$