r1在某種環(huán)境下貓頭鷹的主要食物是田鼠，設(shè)田鼠的年平均增長(zhǎng)率為 ，貓頭鷹的存1a在引起的田鼠增長(zhǎng)率的減少與貓頭鷹的數(shù)量成正比，比例系數(shù)為r

1 ；貓頭鷹的年平均減少率為a 2

；田鼠的存在引起的貓頭鷹減少率的增加與田鼠的數(shù)量成正比，比例系數(shù)為2 。建立差分方程模型描述田鼠和貓頭鷹共處時(shí)的數(shù)量變化規(guī)律，對(duì)以下情況作圖給出50年的變化過(guò)程。設(shè)r1

0.2,r2

0.3,a1

0.001,a2

0.002,,0和50只貓頭鷹。1 2 1 2 100只田鼠200aa適當(dāng)改變參數(shù) 1 2初始值同上）求差分方程的平衡點(diǎn)，它們穩(wěn)定嗎？k解：記第k代田鼠數(shù)量為xk，第k代貓頭鷹數(shù)量為y，則可列出下列方程：kx =x+(r-y*a)*xK+1 K 1 K 1 KK+1=yK+(-r2+xK*a2)*yK運(yùn)用matlab計(jì)算，程序如下：functionz=disiti(x0,y0,a1,a2,r1,r2)x=x0;y=y0;fork=1:49x(k+1)=x(k)+(r1-y(k)*a1)*x(k);y(k+1)=y(k)+(-r2+x(k)*a2)*y(k);endz=[x',y'];（1）z=disiti(100,50,0.001,0.002,0.2,0.3)plot(1:50,z(:,1));holdon;plot(1:50,z(:,2),'r')得到如下結(jié)果：z=100.000050.0000115.000045.0000132.825041.8500153.831340.4125178.380840.7221206.793043.0336239.252547.9216275.637656.4758315.198370.6668355.963994.0149393.6908132.7422420.1696197.4383421.2459304.1220377.3850469.1057275.8285682.4408142.7576854.181949.3682841.809217.6832672.38369.3300494.44836.5828355.34025.5602253.41645.2632180.20965.3674128.04365.753691.00516.380764.75087.243746.15188.358132.97499.754123.633611.474417.004613.574212.293516.12218.939219.20246.545722.91724.833327.38993.604932.76912.720939.23382.082946.99881.621556.32241.287567.51431.046380.94650.873797.06510.7530116.40510.6733139.60770.6280167.44160.6150200.82690.6364240.86450.7011288.86850.8286346.40291.0587415.31671.4745497.76772.257090080070060050040030020010000 5 10 15 20 25 30 35 40 45 50紅線(xiàn)為貓頭鷹數(shù)量曲線(xiàn),藍(lán)線(xiàn)為田鼠曲線(xiàn)(2)z=disiti(100,200,0.001,0.002,0.2,0.3)plot(1:50,z(:,1));holdon;plot(1:50,z(:,2),'r')z=100.0000200.0000100.0000180.0000102.0000162.0000105.8760146.4480111.5459133.5243118.9610123.2551128.0906115.6037138.9010110.5381151.3273108.0844165.2367108.3713180.3771111.6737196.3091118.4584212.3165129.4298227.2997145.5610239.6737168.0647247.3278198.2066247.7713236.7886238.6561283.0909218.8260333.2864189.6595379.1639155.6793409.2388123.1052413.887296.7745391.624478.2302349.935666.5007299.706059.8702249.655556.8973204.652756.6326166.545258.5272135.445462.3054110.666367.871491.256675.251976.267184.563164.865495.990556.3762109.777050.2865126.212146.2412145.618344.0412168.328743.6553194.646145.2556224.766549.2965258.639556.6680295.710968.9807334.454789.0832371.5513121.9469400.5521175.9818410.1726264.1671383.8530401.6251306.4586589.4677187.1029773.922279.7204831.351790080070060050040030020010000 5 10 15 20 25 30 35 40 45 50紅線(xiàn)為貓頭鷹數(shù)量曲線(xiàn),藍(lán)線(xiàn)為田鼠曲線(xiàn)(3)當(dāng)a1,a2分別取0.002,0.002時(shí)，得到如下圖像：4504003503002502001501005000 5 10 15 20 25 30 35 40 45 50a1,a2(4)令xK=xK+1=x; 解方程得到如下結(jié)果：x=150y=200經(jīng)matlab驗(yàn)證如下：z=disiti(150,200,0.001,0.002,0.2,0.3)plot(1:50,z(:,1));holdon;plot(1:50,z(:,2),'r')z=1502001502001502001502001502001502001502001502001502001502001502001502001502001502001502001502001502001502001502001502001502001502001502001502001502001502001502001502001502001502001502001502001502001502001502001502001502001502001502001502001502001502001502001502001502001502001502001502001502001502002001951901851801751701651601551500 5 10 15 20 25 30 35 40 45 50由此可知：平衡點(diǎn)為y=200研究將鹿群放入草場(chǎng)后草和鹿兩種群的相互作用。草的生長(zhǎng)遵從Logistic規(guī)律，年固有增長(zhǎng)率0.，最大密度為300（密度單位，在草最茂盛時(shí)每只鹿每年可吃掉1.（密度單位0.9，而草的存在可使鹿的死亡得以補(bǔ)償，在草最茂盛時(shí)補(bǔ)償率為。作一些簡(jiǎn)化假設(shè)，用差分方程模型描述草和鹿兩種群數(shù)量的變化過(guò)程，就以下情況進(jìn)行討論：10010003000的草場(chǎng)兩種情況。適當(dāng)改變參數(shù)，觀察變化趨勢(shì)。解：設(shè)1．草獨(dú)立生存，獨(dú)立生存規(guī)律遵從Logistic規(guī)律；2．草場(chǎng)上除了鹿以外，沒(méi)有其他以草為食的生物；鹿無(wú)法獨(dú)立生存。沒(méi)有草的情況下，鹿的年死亡率一定；假定草對(duì)鹿的補(bǔ)償率是草場(chǎng)密度的線(xiàn)性函數(shù)；每只鹿每年的食草能力是草場(chǎng)密度的線(xiàn)性函數(shù)。r，N，d，a，b；k＋1，鹿的數(shù)量為 ，第k年草的密度為 ，鹿的數(shù)量為 。草獨(dú)立生存時(shí)，按照Logistic規(guī)律增長(zhǎng)，則此時(shí)草的增長(zhǎng)差分模型為，但是由于鹿對(duì)草的捕食作用，草的數(shù)量會(huì)減少，則滿(mǎn)足如下方程：（ ）（1）鹿離開(kāi)草無(wú)法獨(dú)立生存，因此鹿獨(dú)立生存時(shí)的模型為存在會(huì)使得鹿的死亡率得到補(bǔ)償，則滿(mǎn)足如下差分方程：

，但是草的（ ）（2）另外，記初始狀態(tài)鹿的數(shù)量為 ，草場(chǎng)密度初值為 。各個(gè)參數(shù)值為：， ， ， ，利用MATLAB編程序分析計(jì)算該差分方程模型，源程序如下：diwutidiwuti-LogisticB=diwuti(x0,y0,r,N,b,a,d,n)diwuti-Logisticx(1)x0; y(1)y0; fork=1:n;x(k+1)=x(k)+r*(1-x(k)/N)*x(k)-a*x(k)*y(k)/N;y(k+1)=y(k)+(-d+b*x(k)/N)*y(k);endB=[x;y];%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%clearallC1=diwuti(1000,100,0.8,3000,1.5,1.6,0.9,50);C2=diwuti(3000,100,0.8,3000,1.5,1.6,0.9,50);k=0:50;plot(k,C1(1,:),'b',k,C1(2,:),'b',k,C2(1,:),'r',k,C2(2,:),'r',),...axis([05003000]);xlabel('時(shí)間/年')ylabel('種群量/草場(chǎng)：?jiǎn)挝幻芏?，鹿：頭')title('圖1.草和鹿兩種群數(shù)量變化對(duì)比曲線(xiàn)')gtext('x0=1000')gtext('x0=3000')gtext(gtext('鹿群數(shù)量')10010003000（1）：1000300040-50量將達(dá)到穩(wěn)定。MatLab衡點(diǎn)為（1800，600）。

，即兩種群數(shù)量的平為進(jìn)一步驗(yàn)證此結(jié)論，下面通過(guò)改變相關(guān)參數(shù)，研究?jī)煞N群變化情況，找到影響平衡點(diǎn)的因素：改變草場(chǎng)密度初始值；2響。改變鹿的數(shù)量初值2理論上平衡值。但是，我們可以看到，y0=2000（紫色曲線(xiàn)），5－15的種群就要滅絕。同樣道理，草場(chǎng)的密度也存在一個(gè)最小量的域值，低于這個(gè)閾值，草也將滅絕。的數(shù)量存在上限。N，畫(huà)圖比較結(jié)果；4N平衡點(diǎn)兩種群的數(shù)量就越小。改變鹿群獨(dú)立生存時(shí)的死亡率5.15.2鹿單獨(dú)生存的死亡率越大，則兩種群數(shù)量達(dá)到平衡點(diǎn)的時(shí)間越短；相反，鹿單獨(dú)生存的死亡率越小，則兩種群數(shù)量達(dá)到平衡點(diǎn)的時(shí)間越長(zhǎng)（甚至有可能會(huì)出現(xiàn)分叉、混沌）。草場(chǎng)密度對(duì)鹿數(shù)量的補(bǔ)償作用變化（b）bbb點(diǎn)，出現(xiàn)多值性。Leslie種群年齡結(jié)構(gòu)的差分方程模型已知一種昆蟲(chóng)每?jī)芍墚a(chǎn)卵一次，六周以后死亡（給出了變化過(guò)程的基本規(guī)律。孵化后2周后成熟，平均產(chǎn)卵100個(gè)，四周齡的成蟲(chóng)平均產(chǎn)卵150個(gè)。假設(shè)每個(gè)卵發(fā)育成2周齡成蟲(chóng)的概率為0.0（稱(chēng)為成活率，2周齡成蟲(chóng)發(fā)育成4周齡成蟲(chóng)的概率為0.。假設(shè)開(kāi)始時(shí)，0~2，2~4，4~62周、4周、6周后各種周齡的昆蟲(chóng)數(shù)目；討論這種昆蟲(chóng)各種周齡的昆蟲(chóng)數(shù)目的演變趨勢(shì)：各周齡的昆蟲(chóng)比例是否有一個(gè)穩(wěn)定值k22446。據(jù)題意可列出下列差分方程：x2(k+1)=x1(k)*0.09x3(k+1)=x2(k)*0.2運(yùn)用matlab編寫(xiě)的程序如下：functionz=diliuti(a,r1,r2,n)x(1)=a;y(1)=a;w(1)=a;fork=1:nx(k+1)=y(k)*100+w(k)*150;y(k+1)=x(k)*r1;w(k+1)=y(k)*r2;endz=[x',y',w'];fork=1:n+1m=x(k)+y(k)+w(k)endplot(1:n+1,x);holdonplot(1:n+1,y,'r');holdonplot(1:n+1,w,'k'),grid計(jì)算前三年的結(jié)果為：z=diliuti(100,0.009,0.2,2)m=300m=m=z=1.0e+004*0.01000.01000.01002.50000.00010.00200.30900.02250.0000x1042.521.510.501 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3（藍(lán)線(xiàn)為0~2周的蟲(chóng)，紅線(xiàn)為2~4周的蟲(chóng)，黑線(xiàn)為4~6周的蟲(chóng)）其中，m表示三個(gè)不同生長(zhǎng)周期的蟲(chóng)的總數(shù)，可見(jiàn)蟲(chóng)并未滅絕。當(dāng)年份足夠長(zhǎng)時(shí)，可觀察到各年齡段蟲(chóng)的數(shù)量變化：>>z=diliuti(100,0.009,0.2,20)m=300m=m=m=2.2600e+004m=m=m=m=m=m=m=m=m=2.7132e+004m=m=m=m=m=m=m=m=z=1.0e+004*0.01000.01000.01002.50000.00010.00200.30900.02250.00002.25270.00280.00450.95310.02030.00062.11090.00860.00411.46600.01900.00172.15710.01320.00381.88930.01940.00262.33720.01700.00392.28280.02100.00342.61360.02050.00422.68560.02350.00412.96860.02420.00473.12270.02670.00483.39690.02810.00533.61200.03060.00563.90030.03250.00614.16790.03510.00654.48550.03750.00704.80420.04040.0075x10454.543.532.521.510.500 5 10 15 20 25由此可見(jiàn)，0~2周的蟲(chóng)的數(shù)量急劇增多，2~4周的蟲(chóng)的數(shù)量也增多，而4~6周的蟲(chóng)的數(shù)量相對(duì)很少。三者并無(wú)太多比例關(guān)系。最終整個(gè)種群數(shù)量增多。當(dāng)使用殺蟲(chóng)劑時(shí)：z=diliuti(100,0.0045,0.1,20)m=300m=2.5010e+004m=1.65