Frontiers in Computing and Intelligent Systems ISSN: 2832-6024 | Vol. 9, No. 1, 2024 56 A Delayed Mathematical Model of Population Pressure on the Atmospheric Level of Carbon Dioxide Gas Tianchen Yao * Faculty of Engineering, University of New South Wales, Sydney, 2050, Australia * Corresponding author Email: yaotianchen639@163.com Abstract: Effective control of carbon dioxide and methane emissions is the key to controlling global warming. For this reason, a delayed mathematical model of population pressure on the atmospheric level of carbon dioxide gas is put forward. The impact of the time lag on the model and exhibition of a Hopf bifurcation at the threshold of the time lag is analyzed in compliance with the allocation of the roots of the appropriate characteristic equation. Besides, the direction and stability of the Hopf bifurcation are determined by drawing support from the center manifold method. Eventually, computer numerical calculations are conducted to validate the correctness of our obtained analytical findings. Keywords: Carbon Dioxide Gas; Delay; Hopf Bifurcation; Mathematical Model; Population Pressure. 1. Introduction With the successive exploitation of science and technology industry, carbon dioxide emissions in the globe are also gradually increasing. With the successive expansion of carbon dioxide emissions, global warming has brought about the variability of the marine climate. Subsequently, many natural catastrophes arise, such as permanent glacier melting, sea level rising, frequent extreme weather and changes in ecosystem structure. In line with The Global Energy Review: Carbon Dioxide Emissions in 2021 released by The International Energy Agency (IEA), carbon dioxide emissions in the global energy sector exploded to 36.3 billion tons, increasing 6% year-on-year and exceeding the level before the outbreak of COVID-19 and setting a record [1]. Accordingly, in the context of tighter resource and environmental constraints, climate change caused by greenhouse gas emissions such as methane, nitrous oxide and ozone, especially carbon dioxide, has become the focus of attention of all sectors of the world [2]. In order to fight against the negative impacts caused by climate change, the world has made efforts to take various measures in all aspects. As early as May 9, 1992, the United Nations General Assembly adopted the Framework Convention on climate change. As of June 2016, 197 countries have adopted the Framework Convention on climate change, covering more than 65% of the total carbon emissions [3]. In 2015, the climate change conference held in Paris established a national independent contribution mechanism, requiring parties to refer to national conditions and propose emission reduction targets to address climate change in a ''bottom-up'' manner [4, 5]. After forwards, in 2019, the 49th plenary session of the Intergovernmental Panel on Climate Change (IPCC) adopted the improvement plan for the 2006 IPCC national inventory guidelines (IPCC, 2019), which clearly added the content of verifying the emission list based on atmospheric concentration and combined with the ''top-down'' (i.e. atmospheric inversion) method. In particular, the Chinese government has clearly put forward the carbon crest value by 2030 and the carbon neutralization target by 2060, from relative emission decline to absolute emission decline and zero emissions [6]. In recent years, some researchers have begun to study how to control carbon dioxide emissions from the perspective of system dynamics. Tennakone [7] revealed that severe deforestation could lead to a rapid increase of the carbon dioxide through constructing a biomass-carbon dioxide mathematical model and analyzing the stability of the constructed model. Alexiadis [8] explored the interplay between global warming and human activities with the help of control theory, and a feedback model is developed to analyze the impact of the carbon dioxide emission because of human activity on the global temperature. Subsequently, Dubey et al. [9] studied the connection between forestry resources and industrialization on account of population and population pressure by analyzing the dynamics of a nonlinear mathematical model. Then, Caetano et al. [10] presented some suggestions to cut down the carbon dioxide emissions with the aid of optimal control theory and by constructing a mathematical model. Misra and Verma [11] explored the influence of human population and forest biomass on carbon dioxide in the atmosphere through establishing a mathematical model. Sundar et al. [12] investigated the reduction of carbon dioxide by utilizing suitable absorbents around the source of carbon dioxide emission. Very recently, considering the interplay among carbon dioxide, human population, population pressure and forest biomass, Misra and Jha [13] formulated a mathematical model about population pressure on carbon dioxide as below on the foundation of the work by Misra and Verma [11]: 0 1 1 0 2 1 ( ) ( ) ( ) ( ) ( ), ( ) ( ) ( ) 1 ( ) ( ) ( ) ( ), ( ) ( ) ( ), ( ) ( ) ( ) 1 ( ) ( ) ( ) ( ). dX t Q N t X t X t F t dt dN t N t sN t X t N t N t F t dt L dP t N t P t dt dF t F t F t N t F t P t F t dt M            = + − −    = − − +      = −     = − − −     (1) Where, ( )X t , ( )N t , ( )P t and ( )F t denote the carbon dioxide consistence in the atmosphere, the human population, the population pressure and the forest biomass at 57 time t . 0Q denotes the natural emission rate of carbon dioxide;  denotes the anthropogenic emission rate coefficient of carbon dioxide;  is the natural depletion rate coefficient of carbon dioxide; 1 denotes the depletion rate coefficient of carbon dioxide due to forestry biomass; s and  are the intrinsic growth rates of the human population and forest biomass, respectively; L and M are the carrying capacities of human population and forest biomass, respectively;  denotes the reduction rate coefficient of the human population on account of adverse effects of carbon dioxide; 1 denotes the increasing rate coefficient of the population pressure due to the increase in the human population; 0 denotes the natural depletion rate coefficient of the population pressure; 1 denotes the diminution rate coefficient of the carrying capacity of forest biomass because of the population pressure;  denotes the depletion rate coefficient of forest because of the human population;  denotes the increasing rate coefficient of the human population because of forest biomass. Obviously, the model (1) assumed that the growth of the human population gets favored by the forest biomass instantaneously. However, as is known to all that it is necessary to take some time to clear land for agriculture and roads etc. Additionally, in order to describe dynamics of mathematical models replying on the past history of the models, it is often needful to introduce time delays into the models [14-17]. Moreover, periodic solutions can appear due to the Hopf bifurcation in mathematical models with time delays [18-21]. Meng and Li [18] studied the bifurcating periodic solutions of a delayed phytoplankton-zooplankton system with Allee effect and linear harvesting. Bentout et al. [19] analyzed the Hopf bifurcation of a double age dependence epidemic model with two delays. Yang et al. [20] established a diffusive predator-prey model with time delay, and they revealed that time delay could make coexisting equilibrium lose stability and give rise to bifurcating periodic solution when it increases through a determinate threshold. Li et al. [21] delved into the Hopf bifurcation of a novel high- dimensional delayed fractional neural network model. Xu et al. [22] established a series of adequate conditions assuring the stability and the emergence of Hopf bifurcation for fractional-order competitive web site model of the Internet. There is also some work about Hopf bifurcation and periodic solutions in other fields [23-26]. Motivated by the work above and inspired by the fact that when the model (1) exhibits a Hopf bifurcation, the carbon dioxide concentration in the atmosphere will be out of control, we consider the model with a time lag as below: 0 1 1 0 2 1 ( ) ( ) ( ) ( ) ( ), ( ) ( ) ( ) 1 ( ) ( ) ( ) ( ), ( ) ( ) ( ), ( ) ( ) ( ) 1 ( ) ( ) ( ) ( ). dX t Q N t X t X t F t dt dN t N t sN t X t N t N t F t dt L dP t N t P t dt dF t F t F t N t F t P t F t dt M                = + − −    = − − + − −      = −     = − − − − −     (2) Where,  is the time lag due to the interval that the growth of the human population needs to get favored by the forest biomass. The remainder of the current paper is outlined as below. The impact of the time delay on the model and exhibition of Hopf bifurcation are explored in Section 2. Explicit formula direction and stability of the Hopf bifurcation are conducted in Section 3. Numerical simulating analysis is demonstrated in Section 4. A conclusion is drawn in Section 5. 2. Exhibition of Hopf Bifurcation In the present section, we shall discuss the impact of the time lag on the model (2), and establish sufficient conditions for exhibition of Hopf bifurcation through analyzing allocation of the roots of the appropriate characteristic equation of the model (2). The method used in this section is common and simple, which is relatively intuitive. If 0s Q  and 0( )L s Q u s L      −  + , then the model (2) exhibits an interior equilibrium * * * * *( , , , )U X N P F , where 0 * * 1 * Q N X F    + = + , * 1 *P A N= , 2 2 * 1 * 0 * 1 * 0 B F B F B N C F C + + = + , and 1 1 0 A   = and *F is the positive root of Eq.(3) as below 3 2 3 * 2 * 1 * 0 0f F f F f F f+ + + = (3) Where 0 0 0f B uC= − , 1 1 1 0 1 3 0 1f A B B A C uC = + + − , 2 1 1 1 2 3 1f A B B A C = + + , 3 1 1 2f A B= , 0 0B s Q = − , 1 1B s = + , 2 1B = , 0 2C A = + , 1 1 2C A= , 2 s A L = , 3 u A M = . The characteristic equation at * * * * *( , , , )U X N P F corresponding to the model (2) can be obtained as below 4 3 2 3 2 1 0 3 2 3 2 1 0 2 2 2 1 0 ( ) ( ) 0, F F F F G G G G e H H H e            − − + + + + + + + + + + + = (4) with 0 0 44 21 11 22 1 14 21 43( )F f f f f f f f  = − + , 1 0 11 22 11 44 22 44 11 22 44 21 0 44( ) ( )F f f f f f f f f f f f  = + + − − − , 2 11 22 11 44 22 44 0 11 22 44 21( )F f f f f f f f f f f = + + − + + − , 3 0 11 22 44F f f f= − − − , 1 0 21 44 14 42 0 11 22 44 44 22( ) ( )G f g f g f f g f g  = − − + , 1 0 11 22 44 44 22 0 11 22 44( )( ) ( )G f f g f g f g g = − + + + , 2 22 44 44 22 22 44 0 11( )( )G f g f g g g f= + − + − , 3 22 44( )G g g= − + , 0 0 11 24 42 22 44( )H f g g g g= − , 1 0 11 22 44 24 42( )( )H f g g g g= − − , 2 22 44 24 42H g g g g= − , And 11 1 *f F = − − , 14 1 *f X= − , 21 *f N= − , 58 * 22 * 2sN f s X L = − − , 43 1 *2f F= − , * 44 2uF f u M = − , 22 *g F= , 24 *g N= , 42 *g F= − , 44 *g N= − . Multiplying by e , Eq. (4) yields 3 2 3 2 1 0 4 3 2 3 2 1 0 2 2 1 0 ( ) ( ) 0. G G G G F F F F e H H H e            − + + + + + + + + + + + = (5) For 0 = , one has 4 3 2 3 2 1 0 0J J J J   + + + + = (6) where 0 0 0 0J G F H= + + , 1 1 1 1J G F H= + + , 2 2 2 2J G F H= + + , 3 3 3J G F= + . Lemma 1[13]. If 2 1 2 3 1 3 0( )J J J J J J−  , then the model (2) is locally asymptotically stable when 0 = . For 0  , putting i = ( 0  ) into Eq. (5), then we get 1 2 3 4 5 6 ( )sin( ) ( ) cos( ) ( ), ( ) cos( ) ( )sin( ) ( ), K K K K K K           + =  + = (7) with 3 1 1 1 3( ) ( )K H F F  = − + , 4 2 2 2 2 0 0( ) ( )K H F F H  = − + + + , 2 3 2 0( )K G G = − , 3 4 1 1 3( ) ( )K H F F  = + − , 4 2 5 2 2 0 0( ) ( )K H F F H  = + − + − , 3 6 3 1( )K G G  = − . Then, we obtain 1 0 ( ) cos( ) ( )     =  (8) 2 0 ( ) sin( ) ( )     =  (9) where 8 2 6 2 2 4 0 3 2 2 0 2 1 3 2 2 2 2 2 1 0 2 0 2 1 0 0 ( ) ( 2 ) ( 2 2 ) ( 2 2 ) , F F F F H F F F F F H H H F H       = + − + + − − + − + − + − 6 4 1 2 3 3 1 3 2 2 2 0 3 1 1 2 2 0 0 0 2 2 1 1 1 0 0 0 ( ) ( ) ( ( ) ( )) ( ( ) ( ) ( )) ( ), G G F G F G F H G G H F G F H G F H G H F G F H      = − + − − − − − + − + − + − + − 7 5 3 3 2 3 1 3 2 2 3 1 2 2 3 0 0 2 1 1 0 3 0 1 1 1 0 0 ( ) ( ( )) ( ( ) ( ) ( ) ) ( ( ) ( )) . G G F G G F H G F H G F H G H F G F G F H G H F       = + − − + + + − + − + − + + − + Further, the algebraic equation regarding  can be get 2 2 2 0 1 2( ) ( ) ( ) 0   − − = . (10) Clearly, one can get all the roots of Eq. (10) once the parameters of the model (2) are determined. Thus, we suppose that Eq. (10) has positive root 0 . Then Eq. (5) has roots 0i .For 0 , on the foundation of Eq. (8), one has 1 0 0 0 0 0 ( )1 2 arccos ( ) i i        =  +    , 0,1,2,i =  . (11) Denote 0 min{ }i = , 0,1,2,i =  . (12) Differentiating Eq.(6) about  ς, then 1 4 3 ( ) ( ) d d      −    = − −    (13) with 5 4 3 2 3 2 3 3 2 1 0 2 1 0( ) ( ) ( )F F F F e H H H e          − = + + + + − + + , 2 3 2 4 3 2 1 3 2 1 2 1( ) 3 2 (4 3 2 ) (2 )G G G F F F e H H e        − = + + + + + + + + . Then 0 1 * 4 0 * 3 0 ( ) Re ( ) i d d      − =    =    (14) Where * 5 3 3 0 0 2 2 0 0 0 0 0 0 2 4 2 1 1 0 3 0 0 0 3 5 2 2 0 0 0 0 0 0 0 2 4 2 1 1 0 3 0 0 0 ( ) (( ( ) ( ) )sin (( ) ) cos ) ((( ) ( ) ) cos (( ) )sin ) , F H F H F H F F H F H F H F                     = − + + + + − − + − − + + + + − , * 5 3 4 0 0 2 2 0 0 0 0 0 0 2 4 1 1 0 3 0 0 0 2 2 1 3 0 1 1 3 0 0 0 3 2 2 0 0 0 0 3 5 2 2 0 0 0 0 0 0 0 2 4 1 1 0 0 0 0 2 0 1 ( ) (( ( ) ( ) )sin (( ) ) cos ) ( 3 ( 3 )cos ((2 2 ) 4 )sin ) ((( ) ( ) ) cos (( ) )sin ) (2 ( F H F H F H F G G F H F F H F H H F F H G F                              = − + + + + − −  − + + − − − − + − − + + + + −  + 2 1 3 0 0 0 3 2 2 0 0 0 0 3 )sin ((2 2 ) 4 )cos ). H F F H        − − + + − 59 Apparently, if * 4 0( ) 0  then we have   0 1 Re / 0 i d d     − =  . On the foundation of the Hop bifurcation theorem in [27], one can get the results as below. Theorem 1. If 0s Q  and 0( )L s Q u s L      −  + , then the interior equilibrium * * * * *( , , , )U X N P F is locally asymptotically stable whenever 0[0, )  ; while a Hopf bifurcation emerges near the interior equilibrium * * * * *( , , , )U X N P F when 0 = and a family of periodic solutions bifurcate from the interior equilibrium * * * * *( , , , )U X N P F . 3. Direction and Stability of the Periodic Solutions In the current section, we will deduce explicit formulas which can determine the direction and stability of the periodic solutions by using the center manifold method innovated by Hassardet al [27]. The calculation steps of the method we used are complicated. Nevertheless, the results we obtained can determine the direction and stability of the periodic solutions with the6parameters of the model (2) obviously. Define 4([ 1,0],R )C C= − , 0 j = + and Rj . Setting 1 *( ) ( )v t X t X= − , 2 *( ) ( )v t N t N= − , 3 *( ) ( )v t P t P= − , 4 *( ) ( )v t F t F= − and ( / )t t → . Then the model (2) equals ( ) ( ) ( , )j t tv t L v F j v= + (15) Where 4: RjL C → and 4: R RF C → are presented by 0 1 2( ) ( )[ (0) ( 1)]jL j L L   = + + − (16) with 11 14 21 22 1 1 0 43 44 0 0 0 0 0 0 0 f f f f L f f         =  −     , 22 24 2 42 44 0 0 0 0 0 0 0 0 0 0 0 g g L g g      =       , and 0 1 2 4( , ) ( )( , ,0, )TF j j F F F = + (17) where 1 1 4(0) (0)F  = − , 2 2 2 1 2 2 4(0) (0) (0) ( 1) ( 1) s F L     = − − + − − , 2 2 4 1 * 3 4 1 * 4 1 3 4 2 42 (0) (0) (0) (0) (0) ( 1) ( 1) u F F P M            = − − + − + − −    On the foundation of the Riesz representation theorem, one can get ( , )j  for [ 1,0] − satisfying 0 1 ( ) ( , ) ( )jL d j     − =  , 4([ 1,0],R )C  − .(18) Setting 0 1 2( , ) ( )( ( ) ( 1))j j L L    = +  +  + , (19) and ( ) is Dirac delta function. For 4([ 1,0],R )C  − , define 0 1 ( ) , 1 0, ( ) ( ) ( , ) ( ), 0, d dD j d j s          −  −   =   =  (20) and 0, 1 0, ( ) ( ) ( , ), 0. R j F j      −   =  = (21) Thus, system (15) becomes ( ) ( ) ( )t tv t A j v R j v= + . (22) For 1 4 *([0,1], (R ) )C , define * 0 1 ( ) , 0 1, ( ) ( ) ( ,0) ( ), 0,T d s s dsD j s d s s s     −    =   − =  (23) and 0 1 0 ( ), ( ) (0) (0) ( ) ( ) ( )s d d                 =− = = − −  (24) Let 0 0 2 3 4( ) (1, , , ) iT e    =    is the eigenvector of (0)D belonged to 0 0i + and 0 0* * * 4 2 3 4( ) (1, , , ) i sTs Q e    =    s the eigenvector of * (0)D belonged to 0 0i − . Correspondingly, in view of the definitions of (0)D ) and * (0)D , we can obtain 0 0 0 0 0 0 14 21 24 0 11 2 24 21 0 22 22 ( ) ( ) i i i f f g e i f g e f i f g e          − − − + −  = + − − 1 2 3 0 0i      = + , 0 11 22 4 14 i f g f  − −  = , * 0 11 2 21 i f f  +  = − , * * 43 4 3 0 0 f i    = − , 0 0 0 0 4 14 24 4 0 44 44 i i f g e i f g e     +  = − + + . From Eq. (24), one has 60 0 0 * * * 2 2 3 3 4 4 * * 0 2 22 2 42 4 * * 1 4 24 2 44 4 [1 ( ( ) ( ))] , i Q e g g g g   − − = +  +  +  +   +  +  +  such that *, 1  = and *, 0  = . Then, we can acquire the explicit expressions of 20h , 11h , 02h and 21h based on the method used in [28, 29]: * * 20 0 201 2 202 4 2042 ( )h Q h h h= + + , * * 11 0 111 2 112 4 114( )h Q h h h= + + , * * 02 0 021 2 022 4 0242 ( )h Q h h h= + + , * * 21 0 211 2 212 4 2142 ( )h Q h h h= + + , with 201 4h = −  , 0 022 202 2 2 2 4 is h e L    − = −  −  +   , 0 022 204 1 * 3 4 1 * 4 2 42 iu h F P e M     −  = −   − +  −      , 111 4 4( )h = −  + , 112 2 2 2 2 2 4 2 4 2 ( ) ( ) s h L  = −   −  + +   +  , 114 1 * 3 4 3 4 1 * 4 4 2 4 2 42 ( ) 2 ( ) u h F P M      = −   +  − +   −   +     , 021 4h = −  , 0 022 022 2 2 2 4 is h e L   = −  −  +   , 0 022 024 1 * 3 4 1 * 4 2 42 iu h F P e M       = −   − +  −      , (1) (1) (4) (4) 211 11 4 20 4 11 20 1 1 (0) (0) (0) (0) 2 2 h W W W W   = −  +  + +    , 0 0 0 0 0 0 0 0 (2) (2) 212 11 2 20 2 (2) (2) (2) (2) 11 2 20 2 11 20 (2) (2) 11 4 20 4 (4) (4) 11 2 20 2 (2 (0) (0) ) 1 1 (0) (0) (0) (0) 2 2 1 ( 1) ( 1) 2 1 ( 1) ( 1) , 2 i i i i s h W W L W W W W W e W e W e W e            − − = −  +    −  +  + +      + −  + −       + −  + −     ( ) 0 0 0 0 0 0 (3) (3) (4) (4) 214 1 * 11 4 20 4 11 3 20 3 (4) (4) 2 1 * 11 4 20 4 1 3 4 3 4 4 (2) (2) 11 4 20 4 (4) ( 11 2 20 1 1 2 (0) (0) (0) (0) 2 2 2 (0) (0) ( 2 ) 1 ( 1) ( 1) 2 1 ( 1) 2 i i i h F W W W W u P W W M W e W e W e W            − −   = −  +  +  +       − +  +  −   +         − −  + −     − −  + 0 04) 2( 1) , i e    −     with 0 0220 02 20 1 0 0 0 0 ( ) (0) (0) 3 iih ih W e        =  +  + , (25) 11 11 11 2 0 0 0 0 ( ) (0) (0) ih ih W      = −  +  + , (26) and 4 1 11 12 13 14( , , , ) RT =      and 4 2 21 22 23 24( , , , ) RT =      are denoted as below: 0 0 0 0 0 0 0 0 201 14 2 2 202 0 22 22 24 11 1 0 020 2 2 204 42 43 0 44 0 2 02 0 2 0 2 i i i i h f h i f g e g e i h g e f i g e               − − − − − − −  = − + − − − , 0 0 0 0 0 11 201 14 2 21 202 24 12 0 020 2 204 43 0 44 2 0 02 0 0 2 0 0 2 i i i f h f f h g e i h f i g e         − − − −  = + − − , 0 0 0 0 0 0 0 0 0 11 201 14 2 2 21 0 22 22 202 24 13 120 2 2 42 204 0 44 2 22 0 0 0 0 2 i i i i i f h f f i f g e h g e g e h i g e              − − − − − − − −  = − − − , 0 0 0 0 0 11 201 2 21 0 22 22 0 0 202 14 120 2 42 43 204 2 0 2 22 0 0 0 0 i i i f h f i f g e i h g e f h           − − − − − +  = − − − , 61 111 14 112 22 22 24 21 1 011 114 42 43 44 44 0 02 0 0 h f h f g g h g f f g    +  = − − + , 11 111 14 21 112 24 22 011 114 43 44 44 0 02 0 0 0 0 f h f f h g h f f g   = − − + , 11 111 14 21 22 22 112 24 23 111 42 114 44 44 2 0 0 0 0 f h f f f g h g g h f g   +  = −  + , 11 111 21 22 22 112 24 1 011 42 43 114 0 02 0 0 0 f h f f g h g f h    +  = − − , with 0 0 0 0 0 0 0 0 0 11 14 2 2 21 0 22 22 24 20 1 0 0 2 2 42 43 0 44 2 0 2 0 0 2 0 0 2 i i i i i f f f i f g e g e i g e f i g e                − − − − − − − −  = − + − − − , 11 14 21 22 22 24 11 1 0 42 43 44 44 0 0 0 0 0 f f f f g g g f f g    +  = − + . Accordingly, we have 2 2 02 21 1 11 0 0 | | (0) 11 20 2 | | 2 3 2 h hi C h h h     = − − +    , 1 1 ' 0 Re{ (0)} Re{ ( )} C    = − , 2 12Re{ (0)}C = . Conclusively, we can acquire the theorem depicting properties of the Hopf bifurcation at 0 = stated as below. Theorem 2. Supposing that 1 0  ( 1 0  ), then Hopf bifurcation emerging at 0 = is supercritical (subscriptical); supposing that 2 0  ( 2 0  ), then the periodic solutions bifurcating from the interior equilibrium * * * * *( , , , )U X N P F are stable (unstable). 4. Numerical Simulations In the current section, we will execute some numerical calculations to validate the effectiveness of obtained theoretical results by using MATLAB software package 7.0 and the same method used in [28]. We choose some same values of parameters in the model (1) by Misra and Jha [13], and give attention to the sufficient requirements for the emergence of Hopf bifurcation in the present paper. Setting 0 1Q = , 0.05 = , 0.003 = , 1 0.0001 = , 0.01s = , 1000L = , 0.00001 = , 0.077 = , 0.0002 = , 1 0.0002 = , 0 0.005 = , 0.2u = , 2000M = , 1 0.00002 = , the model (2) equals: ( ) 1 0.05 ( ) 0.003 ( ) 0.0001 ( ) ( ), ( ) ( ) 0.01 ( ) 1 0.00001 ( ) ( ) 0.0000154 ( ) ( ), 1000 ( ) 0.0002 ( ) 0.005 ( ), ( ) ( ) 0.2 ( ) 1 0.0002 ( ) ( ) 0.000 2000 dX t N t X t X t F t dt dN t N t N t X t N t N t F t dt dP t N t P t dt dF t F t F t N t F t dt     = + − −   = − − + − −    = −   = − − − − −    202 ( ) ( ).P t F t            (27) Evidently, 00.00003 0.00001s Q =  = and 0( ) 0.2 0.0075 L s Q u s L      − =  = + . Conclusively, the 62 conditions for the exhibition of an interior equilibrium of the model (28) hold. Further, With the aid of MATLAB software package, we get the unique interior equilibrium *(903.4307,456.4565,18.2583,233.6929)U . Correspondingly, in line with Lemma 1, the model (28) is locally asymptotically steady when 0 = . For 0  , by some calculations, we obtain ̟ 0 3.2683 = , 0 20.0629 = and ' 0( ) 3.9658 004 1.6086 005e i e  = − − + − . Accordingly, adequate conditions for the exhibition of Hopf bifurcation are satisfied. Setting 019.5945 [0, ) =  , as can be seen from Figure 1 that the values of the four states in the model (28) tend to the interior equilibrium *(903.4307,456.4565,18.2583,233.6929)U gradually. This suggests that the model (28) is locally asymptotically steady, which can be also described by Figure 2. Next, we fix 021.3107 =  . In this case, the model (28) is bereft of its stability and a Hopf bifurcation emerges near 0 20.0629 = , and a cluster of periodic solutions occur around *(903.4307,456.4565,18.2583,233.6929)U . As shown in Figure 3, the values of the four states in the model (28) exhibit periodic oscillations near the interior equilibrium *(903.4307,456.4565,18.2583,233.6929)U . This phenomenon can be also described in Figure 4. These results are consistent with the Theorem 1. Figure 1. Time plots of the model (28) for 019.5945 =  Figure 2. Phase plots of the model (28) for 019.5945 =  63 Figure 3. Time plots of the model (28) for 021.3107 =  Figure 4. Phase plots of the model (28) for 021.3107 =  5. Conclusion In the current paper, a delayed mathematical model of population pressure on the atmospheric level of carbon dioxide gas is proposed by introducing the time delay on account of the interval that the growth of the human population needs to get favored by the forest biomass into the model innovated by Misra and Jha. The main aim of the current paper is to analyze the impact of the time delay on our proposed model. A sequence of adequate criteria for the local stability of the model and emergence of Hopf bifurcation are derived with the help of the eigenvalue method. Direction and stability of the Hopf bifurcation are probed by utilizing the center manifold method. We find that when the value of the 64 time delay on account of the interval that the growth of the human population needs to get favored by the forest biomass is suitable small ( 0[0, )  ), the model is locally asymptotically steady. Under such circumstances, the carbon dioxide consistence in the atmosphere, the human population, the population pressure and the forest biomass gradually tend to interior equilibrium * * * * *( , , , )U X N P F , and the carbon dioxide consistence in the atmosphere can be predicted and controlled. Otherwise, once the value of the time delay on account of the interval that the growth of the human population needs to get favored by the forest biomass exceeds the threshold 0 , then the model loses its stability and the carbon dioxide consistence in the atmosphere, the human population, the population pressure and the forest biomass tend to periodic oscillation, and the carbon dioxide consistence in the atmosphere will be out of control. 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